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A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P 4 T 2 − BVD (polynomial of 4-degree and THINC function of 2-level reconstruction based on BVD algorithm), is formulated as a two-stage spatial reconstruction scheme following the BVD (Boundary Variation Diminishing) principle that minimizes the jumps of the reconstructed values at cell boundaries. In the P 4 T 2 − BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) functions with two-level steepness are used as the candidate reconstruction functions. The final reconstruction function is selected through the two-stage BVD algorithm so as to effectively control both numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P 4 T 2 − BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it produces accurate solutions in long term computation. P 4 T 2 − BVD , as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales.

The direct-forcing immersed boundary is widely adopted to study particle-laden flows. The effective hydrodynamic diameter of the particle is much or less overestimated by the original immersed boundary method, depending on the particle Reynolds number and grid resolution. In this paper, we propose an improved method to dynamically correct the effective hydrodynamic diameter by retracting inward the Lagrangian points to varying distances. The retraction distance is determined by querying a function fitted in this paper. The improved method is tested and validated by several cases, including falling of a spherical particle under gravity, the uniform flow past two stationary particles and the drafting-kissing-tumbling phenomenon of two settling particles. It turns out the improved method not only provides better results for the drag force but also predicts the flow field more accurately. What's more, due to the insensitivity to grid resolution, the improved method is suitable for simulating large-scale fluid–particle systems such as fluidized bed, which are computationally expensive.

The finite-difference time-domain (FDTD) method has been extensively used to simulate varieties of electromagnetic interaction problems. However, because of its Courant–Friedrich–Levy (CFL) condition, the maximum time step size of this method is limited by the minimum size of cell used in the computational domain. So the FDTD method is inefficient to simulate the electromagnetic problems which have very fine structures. To deal with this problem, the Hybrid Implicit Explicit (HIE)-FDTD method is developed. The HIE-FDTD method uses the hybrid implicit explicit difference in the direction with fine structures to avoid the confinement of the fine spatial mesh on the time step size. So this method has much higher computational efficiency than the FDTD method, and is extremely useful for the problems which have fine structures in one direction. In this paper, the basic formulations, time stability condition and dispersion error of the HIE-FDTD method are presented. The implementations of several boundary conditions, including the connect boundary, absorbing boundary and periodic boundary are described, then some applications and important developments of this method are provided. The goal of this paper is to provide an historical overview and future prospects of the HIE-FDTD method.

This paper presents an automatic mesh generation algorithm for body-fitted structured meshes in Computational Fluids Dynamics (CFD) analysis using the Advancing Extraction Method (AEM). The method is applicable to two-dimensional domains with complex geometries, which have the hierarchical tree-like topography with extrusion-like structures (i.e., branches or tributaries) and intrusion-like structures (i.e., peninsula or dikes). With the AEM, the hierarchical levels of sub-domains can be identified, and the block boundary of each sub-domain in convex polygon shape in each level can be extracted in an advancing scheme. In this paper, several examples were used to illustrate the effectiveness and applicability of the proposed algorithm for automatic structured mesh generation, and the implementation of the method.

Solving acoustic equations in the time domain, possibly coupled with the description of flexible structure dynamics, remains attractive as compared to solving the same in the frequency domain: this allows for better consideration of local non-linearities (acoustics/structure), and the boundary integral formulation (also known as BEM) offers an exact description of the infinite acoustic field based on a simple surface mesh (no need for 3D-volume discretization). Some issues remain however: the required memory space and computation time continue to grow rapidly when the number of elements of the surface mesh increases. In the case of a structure with a regular non-slender shape, the computational cost, measured in terms of required memory space, varies by Helmholtz number to the power of 4. This paper illustrates how the accelerating method called NGTD helps overcome this difficulty. This paper shows the applicability of 2 level NGTD to acoustic and vibroacoustic problems described solely by the hypersingular formulation for surfaces. It goes into more detail on some important aspects of the interpolation process and on the memory saving obtained. Implementation within the MOT (“March-On-Time”) ASTRYD code shows the benefits of this method. The memory requirement shows an estimated trend lower than power 1.35 of the number of surface elements.

Motivated by providing well-behaved fully discrete schemes in practice, this paper extends the asymptotic analysis on time integration methods for non-equilibrium radiation diffusion in 2 to space discretizations. Therein studies were carried out on a two-temperature model with Larsen's flux-limited diffusion operator, both the implicitly balanced (IB) and linearly implicit (LI) methods were shown asymptotic-preserving. In this paper, we focus on asymptotic analysis for space discrete schemes in dimensions one and two. First, in construction of the schemes, in contrast to traditional first-order approximations, asymmetric second-order accurate spatial approximations are devised for flux-limiters on boundary, and discrete schemes with second-order accuracy on global spatial domain are acquired consequently. Then by employing formal asymptotic analysis, the first-order asymptotic-preserving property for these schemes and furthermore for the fully discrete schemes is shown. Finally, with the help of manufactured solutions, numerical tests are performed, which demonstrate quantitatively the fully discrete schemes with IB time evolution indeed have the accuracy and asymptotic convergence as theory predicts, hence are well qualified for both non-equilibrium and equilibrium radiation diffusion. Provide AP fully discrete schemes for non-equilibrium radiation diffusion.Propose second order accurate schemes by asymmetric approach for boundary flux-limiter.Show first order AP property of spatially and fully discrete schemes with IB evolution.Devise subtle artificial solutions; verify accuracy and AP property quantitatively.Ideas can be generalized to 3-dimensional problems and higher order implicit schemes.

The work described here is part of a continuing project to develop, analyze, and test some procedures for using discontinuous Galerkin (DG) numerical methods in multi-layer, isopycnic models of ocean circulation. The steps taken in the present paper include the following. (1) Develop an implementation of horizontal viscosity for usage in DG methods for multi-layer models. This step involves a formulation of the local DG method that can be used in the context of barotropic-baroclinic splitting, a widely-used approach to handling the multiple time scales in ocean circulation models. (2) Develop techniques that enable a layered model to exhibit thin layers without computational failures. Layers with negligible thickness can develop in situations that include coastal upwelling, outcropping of surfaces of constant density to the upper boundary of the fluid due to lateral variations in temperature, or intersections of density surfaces with bottom topography. For the sake of DG computations involving thin layers, this paper develops (i) implementations of wind stress, bottom stress, and interfacial shear stress that do not provoke spuriously large velocities, and (ii) a limiter that maintains nonnegative layer thicknesses in DG solutions. (3) Test the above techniques in numerical experiments involving model problems.

This paper presents eigensolution and non-modal analyses for immersed boundary methods (IBMs) based on volume penalization for the linear advection equation. This approach is used to analyze the behavior of flux reconstruction (FR) discretization, including the influence of polynomial order and penalization parameter on numerical errors and stability. Through a semi-discrete analysis, we find that the inclusion of IBM adds additional dissipation without changing significantly the dispersion of the original numerical discretization. This agrees with the physical intuition that in this type of approach, the solid wall is modelled as a porous medium with vanishing viscosity. From a stability point of view, the variation of penalty parameter can be analyzed based on a fully-discrete analysis, which leads to practical guidelines on the selection of penalization parameter. Numerical experiments indicate that the penalization term needs to be increased to damp oscillations inside the solid (i.e. porous region), which leads to undesirable time step restrictions. As an alternative, we propose to include a second-order term in the solid for the no-slip wall boundary condition. Results show that by adding a second-order term we improve the overall accuracy with relaxed time step restriction. This indicates that the optimal value of the penalization parameter and the second-order damping can be carefully chosen to obtain a more accurate scheme. Finally, the approximated relationship between these two parameters is obtained and used as a guideline to select the optimum penalty terms in a Navier-Stokes solver, to simulate flow past a cylinder.

An automatic approach for 3D analysis of acoustic-structure interaction problems is proposed based on the scaled boundary finite element method (SBFEM). The acoustic domain studied in this paper is assumed to be infinite. The infinite acoustic domain is divided into a near field (bounded domain) and a far field (unbounded domain). The acoustic near field contains structures of arbitrary shape, while the far field represents the unbounded acoustic domain. For modelling the wave propagation accurately and efficiently, continued fractions are employed to evaluate the dynamic stiffness and impedance of subdomains in both structural and acoustic domains. The time-domain equations for both structural and acoustic domains can be obtained by introducing auxiliary variables. Via satisfying the boundary conditions on the acoustic-structure interface, the global system of equations for acoustic-structure interaction system can be constructed. Symmetric formulations can also be obtained for this coupled system. Since the SBFEM requires the discretization of only the boundary, the mesh transition on the acoustic-structure interface is easily addressed by the subdivisions of 2D surface elements. Automatic meshing techniques can be incorporated in the proposed approach to generate meshes directly from the input geometrical models. Numerical examples are presented to demonstrate the accuracy, efficiency and potential of the proposed approach for modelling complex 3D acoustic-structure interaction problems.

This paper is concerned with the lattice Boltzmann method for general convection-diffusion equations. For such equations, we develop a multiple-relaxation-time lattice Boltzmann model and show its consistency under the diffusive scaling. The second-order accuracy of the half-way anti-bounce-back scheme accompanying the present MRT model is justified based on an elegant relation of the collision matrix. Using the half-way anti-bounce-back scheme as a central step, we further construct some parameterized single-node second-order schemes for curved boundaries. The accuracy of the proposed model and boundary schemes are numerically validated with several nonlinear convection-diffusion equations.

In this paper we present an efficient and accurate integral equation method to compute the scattering resonances for a subwavelength metallic slit structure. A new boundary integral equation is derived for the scattering problem, and the computation of scattering resonances is reduced to solving the eigenvalues of the corresponding homogeneous formulation over the complex plane. The integral operators are evaluated with high-order precisions by accurate calculations of the Green's functions for the layered medium and accelerated computation of the slit Green's function. The Newton's method is employed for solving the eigenvalues of the boundary integral formulation. We propose an effective strategy for obtaining the initial guesses of scattering resonances by introducing an approximate model for the scattering problem, for which the leading orders of the resonances are derived by asymptotic analysis. Numerical experiments are provided to demonstrate the accuracy, efficiency, and robustness of the method.

In this paper, a high-order compact finite difference algorithm is proposed for the pure streamfunction (vector potential) formulation of the three dimensional steady incompressible Navier–Stokes equations, in which the grid values of the streamfunction (vector potential), its first-order and second-order derivatives are carried as the unknown variables. The numerical boundary schemes are also established for a general set of flow problems with no normal speed on the boundaries. Numerical examples, including a test problem with an analytical solution and three types of lid-driven cubic cavity flow problems, are solved numerically by the newly proposed scheme. The results obtained prove that the present numerical method has the ability to solve the three dimensional incompressible flow with high accuracy.

This paper presents an immersed boundary method (IBM) to model the behaviour of solid–fluid interfaces in incompressible flows on non-conforming, unstructured meshes. Based on a smooth-interface direct forcing formulation, the proposed forcing method directly and implicitly uses the discretised momentum equations to evaluate the source terms required to enforce a no-slip condition on the boundary. The IBM is derived for a finite-volume framework with an implicit pressure-velocity coupling and a colocated variable arrangement, to enable application in complex domains. The novel framework is applied to flows with stationary and moving IBs on both Cartesian and tetrahedral meshes, and the results are compared to and validated with findings reported in the literature. The results demonstrate that the proposed formulation results in an accurate enforcement of the no-slip condition on the IB at every time-step, even for flows with strong transient behaviour and high velocity and pressure gradients. The results are similar to or better than those obtained with other methods that can only be applied on Cartesian meshes. The current method is shown to accurately preserve local continuity in the vicinity of the IB, ensuring local and global mass conservation in addition to satisfying the local no-slip condition. Moreover, the proposed forcing framework shows good results for unstructured meshes, with a similar accuracy as obtained on Cartesian meshes.

This paper develops appropriate boundary conditions for the two-sided fractional diffusion equation, where the usual second derivative in space is replaced by a weighted average of positive (left) and negative (right) fractional derivatives. Mass preserving, reflecting boundary conditions for two-sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary. Stable, consistent explicit and implicit Euler methods are detailed, and steady state solutions are derived. Steady state solutions for two-sided fractional diffusion equations using both Riemann–Liouville and Caputo flux are computed. For Riemann–Liouville flux and reflecting boundary conditions, the steady-state solution is singular at one or both of the end-points. For Caputo flux and reflecting boundary conditions, the steady-state solution is a constant function. Numerical experiments illustrate the convergence of these numerical methods. Finally, the influence of the reflecting boundary on the steady-state behavior subject to both the Riemann–Liouville and Caputo fluxes is discussed.

Local artificial boundary conditions (ABCs) for the numerical simulation of waves have been successfully used for decades (most notably, the boundary conditions due to Engquist & Majda, Bayliss, Gunzburger & Turkel, and Higdon). The basic idea behind these boundary conditions is that they cancel several leading terms in an expansion of the solution. The larger the number of terms canceled, the higher the order of the boundary condition and, in turn, the smaller the reflection error due to truncation of the original unbounded domain by an artificial outer boundary. In practice, however, the use of local ABCs has been limited to low orders (first and second), because higher order boundary conditions involve higher order derivatives of the solution, which may harm well-posedness and cause numerical instabilities. They are also difficult to implement especially in finite elements. A prominent exception is the development of local high order ABCs based on auxiliary variables. In the current paper, we implement high order Bayliss–Turkel ABCs directly — with no auxiliary variables yet no discrete approximation of the constituent high order derivatives either. Instead, we represent the solution at the boundary as an expansion with respect to a continuous basis. For the spherical artificial boundary, the basis consists of eigenfunctions of the Beltrami operator (spherical harmonics), which enable replacing the high order derivatives in the ABCs with powers of the corresponding eigenvalues. The continuous representation at the boundary is coupled to higher order compact finite differences inside the domain by the method of difference potentials (MDP). It maintains high order accuracy even when the boundary is not aligned with the discretization grid.

The paper proposes a novel computational method for simulating diffraction of obliquely incident plane waves by a doubly periodic array of arbitrarily shaped planar delaminations in elastic layered structures. The presented approach is an extension of the boundary integral equation method (BIEM) based on the boundary integral approach. Periodically situated delaminations are simulated via spring boundary conditions (SBCs), which allows considering bridged cracks. A boundary integral equation is formulated for a reference delamination based on SBCs in terms of the Fourier transform of Green's matrix and solved using Galerkin method. Two modifications of the method are proposed: the spectral BIEM applied only for rectangular delaminations and the meshless BIEM with axisymmetric basis functions, which is applied for delaminations of arbitrary shapes. The effectiveness of regular and irregular meshes is studied for the meshless BIEM. Comparison of both modifications of the BIEM is provided for open and bridged cracks. The proposed method allows considering specific wave phenomena such as a band-gap formation or opening of an interface for wave propagation in elastic layered structures with a doubly periodic array of inhomogeneities (e.g. phononic crystals and damaged interfaces).

International audience; Large computation time is widely considered to be the most important issue in scientific research especially in solving structural evolution problems. Recent developments in this domain have shown that the use of non-incremental schemes through Model Order Reduction led to important results in saving time. Yet, the question arises here how to obtain more time-saving. This paper examines an approach based on a collection of significant modes given by Proper Generalized Decomposition (PGD) solution for different time scales in order to save more computation time. The dictionary of the significant modes allows to construct an accurate solution for different characteristic times and different boundary problems compared to the full solution with a relative error rate less than 5% and with a large time saving of order 50 compared to Finite Element Method (FEM). The ability of the approach with respect to cycle time is discussed.

In this paper, a moving reference frame-based gas-kinetic BGK (MRF–BGK) scheme is developed for simulation of viscous flows around arbitrarily moving bodies. Different from most of existing gas-kinetic schemes, the BGK model is now constructed in a general moving reference frame. Due to the non-inertial effects, there will be an additional acceleration term in the Boltzmann equation, which should be carefully defined in order to correctly recover the concerned Navier–Stokes (N–S) equations. This recovery actually serves as a basis for developing the present method. Within the finite volume framework, the BGK model with the inclusion of the acceleration term is locally solved and the numerical fluxes at the cell interface can be evaluated. Through this way, the source (acceleration) term effects can be naturally included into the gas evolution process and the resulting fluxes. Considering that explicit BGK schemes may show low efficiency for certain unsteady applications where the physical time scales are significantly larger than the spatial scales, the dual time-stepping approach together with an implicit BGK scheme is employed. Numerical tests show that the developed MRF–BGK scheme can be well applied for several incompressible and compressible viscous flows. The accuracy is shown to be comparable to the moving grid BGK method, however the mesh movement is avoided in the present method. Also as compared with the immersed boundary BGK method, it is more efficient to deal with high-Reynolds number flows.

We consider the problem of reconstructing unknown inclusions inside a thermal conductor from boundary measurements, which arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the unknown inclusion and gave its rigorous mathematical justification. In this paper, we continue our previous works and provide a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we analyze the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function of an interior transmission problem for the inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map. Our new findings reveal the essence of the reconstruction method. A convergence result for noisy measurement data is also proved. Second, based on the heat layer potential argument, we perform a numerical implementation of the reconstruction method for the homogeneous inclusion case. Numerical results are presented to show the efficiency and stability of the proposed method.

The Bloch mode synthesis (BMS) model-reduction method adapts component mode synthesis techniques to unit-cell problems in order to obtain a reduced-order model that quickly produces band-structure frequencies for any wave vector, or vice versa. Fundamental to BMS is a partitioning of the real-space model into interior and boundary components, and subsequent reduction of the interior via truncated normal mode expansion. In this paper, two enhancements are presented for the BMS method that reduce both computation time and error in band-structure calculations. The first enhancement improves the accuracy of the interior reduction by approximating the participation of the residual modes rather than simply truncating them. The original formulation of BMS includes a modal reduction of the boundary that must be recomputed for every wave vector. This limits computational benefits and prevents the reduced-order model from being useful for the inverse band-structure problem (i.e., the k ( ω ) calculation). The second enhancement is a local boundary reduction that is independent of wave vector and thus does not suffer from the aforementioned limitations.

In this paper we propose a method for the strong imposition of random Dirichlet boundary conditions in the Dynamical Low Rank (DLR) approximation of parabolic PDEs and, in particular, incompressible Navier Stokes equations. We show that the DLR variational principle can be set in the constrained manifold of all S rank random fields with a prescribed value on the boundary, expressed in low rank format, with rank smaller then S . We characterize the tangent space to the constrained manifold by means of a Dual Dynamically Orthogonal (Dual DO) formulation, in which the stochastic modes are kept orthonormal and the deterministic modes satisfy suitable boundary conditions, consistent with the original problem. The Dual DO formulation is also convenient to include the incompressibility constraint, when dealing with incompressible Navier Stokes equations. We show the performance of the proposed Dual DO approximation on two numerical test cases: the classical benchmark of a laminar flow around a cylinder with random inflow velocity, and a biomedical application for simulating blood flow in realistic carotid artery reconstructed from MRI data with random inflow conditions coming from Doppler measurements.

This paper discusses issues in hyperbolic advection–diffusion schemes for high-Reynolds-number boundary-layer problems. Implicit hyperbolic advection–diffusion solvers have been found to encounter significant convergence deterioration for high-Reynolds-number problems with boundary layers. The problems are examined in details for a one-dimensional advection–diffusion model, and resolutions are discussed. One of the major findings is that the relaxation length scale needs to be inversely proportional to the Reynolds number to minimize the truncation error and retain the dissipation of the hyperbolic diffusion scheme. Accurate, robust, and efficient boundary-layer calculations by hyperbolic schemes are demonstrated for advection–diffusion equations in one and two dimensions.

This paper presents a topology optimization method for two dimensional turbulent flow based on the Reynolds-averaged Navier-Stokes (RANS) equations using a level set boundary expression and the immersed boundary method (IBM). In this study, two-equation turbulence models, the k-ϵ and the k-ω, are considered. In our proposed method, the level set method is used for capturing the exact fluid-solid interfaces. Additionally, the no-slip boundary condition along the fluid-solid interfaces is imposed explicitly using the IBM during the optimization process. Based on the information of the exact boundary position, the interpolated velocity and pressure values within the viscous sublayer are estimated using the standard wall function. From the above formulations, we construct a topology optimization method for the total pressure drop minimization problems considering two dimensional turbulent flows, under the frozen turbulence assumption. We provide numerical examples to confirm the validity and utility of the proposed method.

The aim of this paper is to propose a new method to solve the inverse scattering problem in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine the Young modulus of elastic unknown “inclusions”, i.e. not observable solid objects, located in the elastic medium, from partial aperture boundary measurements. This approach combines a time-reversal approach and redatuming - a technique that consists in virtually moving the sensors from the original acquisition location to an arbitrary position - which allows to reduce the size of the inspected medium. Thanks to this strategy, we can recover the properties of the scatterer, this technique being also relatively insensitive to noise in the data.

It is crucial to understand the degradation of implosion performance caused by three-dimensional physics and the implementation of 3-D spherical mesh is challenging. In this paper a mesh generation method based on the cube spherical projection is proposed for 3D implosion problem in Inertial Confinement Fusion (ICF). This kind of mesh has the advantage of good quality, convenient adding radiation source and adaptive moving in the radial direction. Together, a mathematical model of radiation hydrodynamics with ion-electron non-equilibrium is present. The equations of radiation hydrodynamics are solved with the operator splitting method, such that the hydrodynamics equations are solved explicitly making use of the capability of well understood explicit schemes while the radiation diffusion and electron/ion conduction parts are solved implicitly by an eleven-point scheme on the non-orthogonal mesh. We present results from several benchmark problems in order to validate our numerical scheme. And finally, we investigate the development of fourth-order modes and compute the disturbance growth process of hot spot boundary, which shown that it conforms to the physical law.

This paper shows how to obtain boundary formulas for discrete-adjoint-based sensitivities in shape optimization problems. The analysis is carried out for inviscid flows in two-dimensional, unstructured triangular grids. The new formulation agrees with known continuous adjoint boundary formulas for the sensitivities, and the derivation sheds light into the required approximations and the reasons for the equivalence (or lack thereof) between boundary and domain-based adjoint sensitivities. Alternative approximations are also discussed.

In this paper we explore arbitrarily high-order Lagrange collocation-type time-stepping schemes for effectively solving high-dimensional nonlinear Klein–Gordon equations with different boundary conditions. We begin with one-dimensional periodic boundary problems and first formulate an abstract ordinary differential equation (ODE) on a suitable infinity-dimensional function space based on the operator spectrum theory. We then introduce an operator-variation-of-constants formula which is essential for the derivation of our arbitrarily high-order Lagrange collocation-type time-stepping schemes for the nonlinear abstract ODE. The nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix under some suitable smoothness assumptions. With regard to the two dimensional Dirichlet or Neumann boundary problems, our new time-stepping schemes coupled with discrete Fast Sine / Cosine Transformation can be applied to simulate the two-dimensional nonlinear Klein–Gordon equations effectively. All essential features of the methodology are present in one-dimensional and two-dimensional cases, although the schemes to be analysed lend themselves with equal to higher-dimensional case. The numerical simulation is implemented and the numerical results clearly demonstrate the advantage and effectiveness of our new schemes in comparison with the existing numerical methods for solving nonlinear Klein–Gordon equations in the literature.

A multidimensional scheme achieves good resolution of strong and weak shocks irrespective of whether the discontinuities are aligned with or inclined to the grid. However, these schemes are computationally expensive. This paper achieves similar effects by hybridizing two schemes, namely, AUSM and DRLLF and coupling them through a novel shock switch that operates unlike existing switches on the gradient of the Mach number across the cell-interface. The schemes that are hybridized have contrasting properties. The AUSM scheme captures grid-aligned (and strong) shocks crisply but it is not so good for non-grid-aligned weaker shocks, whereas the DRLLF scheme achieves sharp resolution of non-grid-aligned weaker shocks, but is not as good for grid-aligned strong shocks. It is our experience that if conventional shock switches based on variables like density, pressure or Mach number are used to combine the schemes, the desired effect of crisp resolution of grid-aligned and non-grid-aligned discontinuities are not obtained. To circumvent this problem we design a shock switch based for the first time on the gradient of the cell-interface Mach number with very impressive results. Thus the strategy of hybridizing two carefully selected schemes together with the innovative design of the shock switch that couples them, affords a method that produces the effects of a multidimensional scheme with a lower computational cost. It is further seen that hybridization of the AUSM scheme with the recently developed DRLLFV scheme using the present shock switch gives another scheme that provides crisp resolution for both shocks and boundary layers. Merits of the scheme are established through a carefully selected set of numerical experiments.

In this paper an immersed boundary method with semi-implicit fluidsolid coupling for mobile particles of arbitrary density ratio is developed. The new scheme does not require any iterations to balance fluid forces and particle forces at the interface. A new formulation of the particle equations of motion is proposed which not only accounts for the particle itself but also for a Lagrangian layer surrounding the particle surface. Furthermore, it is shown by analytical considerations that the six equations for the linear and angular velocity of the spherical particle decouple which allows their sequential solution. On this basis a new time integration scheme is obtained which is unconditionally stable for all fluidsolid density ratios and enables large time steps, with Courant numbers around unity. The new scheme is extensively validated for various test cases and its convergence is assessed. An appealing issue is that compared to existing immersed boundary methods the new scheme only alters coefficients in the particle equations and the order of the steps, making it easy to implement in present codes with explicit coupling. This substantially extends the field of application of such methods. An extremely efficient semi-implicit IBM for mobile particles of arbitrary density ratio is developed.Unconditional stability is achieved without any iterative coupling to balance fluid and particle forces at the interface.The new scheme only alters coefficients in the particle equations and the sequence of the operations in each time step.Extensive validation for various test cases, including buoyant particles with zero mass.

The Huygens' principle and lacunae can help construct efficient far-field closures for the numerical simulation of unsteady waves propagating over unbounded regions. Those closures can be either standalone or combined with other techniques for the treatment of artificial outer boundaries. A standalone lacunae-based closure can be thought of as a special artificial boundary condition (ABC) that is provably free from any error associated with the domain truncation. If combined with a different type of ABC or a perfectly matched layer (PML), a lacunae-based approach can help remove any long-time deterioration (e.g., instability) that arises at the outer boundary regardless of why it occurs in the first place. A specific difficulty associated with Maxwell's equations of electromagnetism is that in general their solutions do not have classical lacunae and rather have quasi-lacunae. Unlike in the classical case, the field inside the quasi-lacunae is not zero; instead, there is an electrostatic solution driven by the electric charges that accumulate over time. In our previous work [23] , we have shown that quasi-lacunae can also be used for building the far-field closures. However, for achieving a provably non-deteriorating performance over arbitrarily long time intervals, the accumulated charges need to be known ahead of time. The main contribution of the current paper is that we remove this limitation and modify the algorithm in such a way that one can rather avoid the accumulation of charge all together. Accordingly, the field inside the quasi-lacunae becomes equal to zero, which facilitates obtaining the temporally uniform error estimates as in the case of classical lacunae. The performance of the modified algorithm is corroborated by a series of numerical simulations. The range of problems that the new method can address includes important combined formulations, for which the interior subproblem may be non-Huygens', and only the exterior subproblem, i.e., the far field, is Huygens'.

The accuracy of compact finite-difference schemes can be degraded by inconsistent domain or box boundary treatments. A consistent higher-order boundary closure is especially important for block-structured Cartesian AMR solvers, where the computational domain is generally decomposed into a large number of boxes containing a relatively small number of grid points. At each box boundary, a consistent higher-order boundary closure needs to be applied to avoid a reduction of the formal order-of-accuracy of the numerical scheme. This paper presents such a boundary closure for the fifth-order accurate compact finite-difference WENO scheme by Ghosh and Baeder [1]. The accuracy of the new boundary closure is validated by employing the method of manufactured solutions. A comparison of the new compact boundary closure with the original explicit boundary closure demonstrates the improved accuracy for the new compact boundary closure, while the behavior of the scheme across discontinuities appears unaffected. The linear stability analysis results indicate that a linearly stable compact WENO boundary closure is achieved.

A second generation wavelet-based adaptive finite-difference Lattice Boltzmann method (FD-LBM) is developed in this paper. In this approach, the adaptive wavelet collocation method (AWCM) is firstly, to the best of our knowledge, incorporated into the FD-LBM. According to the grid refinement criterion based on the wavelet amplitudes of density distribution functions, an adaptive sparse grid is generated by the omission and addition of collocation points. On the sparse grid, the finite differences are used to approximate the derivatives. To eliminate the special treatments in using the FD-based derivative approximation near boundaries, the immersed boundary method (IBM) is also introduced into FD-LBM. By using the adaptive technique, the adaptive code requires much less grid points as compared to the uniform-mesh code. As a consequence, the computational efficiency can be improved. To justify the proposed method, a series of test cases, including fixed boundary cases and moving boundary cases, are invested. A good agreement between the present results and the data in previous literatures is obtained, which demonstrates the accuracy and effectiveness of the present AWCM-IB-LBM.

The threshold dynamics method developed by Merriman, Bence and Osher (MBO) is an efficient method for simulating the motion by mean curvature flow when the interface is away from the solid boundary. Direct generalization of MBO-type methods to the wetting problem with interfaces intersecting the solid boundary is not easy because solving the heat equation in a general domain with a wetting boundary condition is not as efficient as it is with the original MBO method. The dynamics of the contact point also follows a different law compared with the dynamics of the interface away from the boundary. In this paper, we develop an efficient volume preserving threshold dynamics method for simulating wetting on rough surfaces. This method is based on minimization of the weighted surface area functional over an extended domain that includes the solid phase. The method is simple, stable with O ( N log ź N ) complexity per time step and is not sensitive to the inhomogeneity or roughness of the solid boundary.

The Dirichlet-to-Neumann finite element method (DtN FEM) has proven to be a powerful numerical approach to solve boundary-value problems formulated in exterior domains. However, its application to elastic semi-infinite domains, which frequently arise in geophysical applications, has been rather limited, mainly due to the lack of explicit closed-form expressions for the DtN map. In this paper, we present a DtN FEM procedure for boundary-value problems of elastostatics in semi-infinite domains with axisymmetry about the vertical axis. A semi-spherical artificial boundary is used to truncate the semi-infinite domain and to obtain a bounded computational domain, where a FEM scheme is employed. By using a semi-analytical procedure of solution in the unbounded residual domain lying outside the artificial boundary, the exact nonlocal boundary conditions provided by the DtN map are numerically approximated and efficiently coupled with the FEM scheme. Numerical results are provided to demonstrate the effectiveness and accuracy of the proposed method. A numerical method for elasticity in axisymmetric semi-infinite domains is proposed.The method couples finite elements with Dirichlet-to-Neumann boundary conditions.The lack of an explicit closed-form expression for the DtN map needs to be overcome.This is done by using a procedure that combines analytical and numerical techniques.The numerical experiments confirm the effectiveness and accuracy of the method.

In this paper we develop a new technique, called \textit{state redistribution}, that allows the use of explicit time stepping when approximating solutions to hyperbolic conservation laws on embedded boundary grids. State redistribution is a postprocessing technique applied after each time step or stage of the base finite volume scheme, using a time step that is proportional to the volume of the full cells. The idea is to stabilize the cut cells by temporarily merging them into larger, possibly overlapping neighborhoods, then replacing the cut cell values with a stabilized value that maintains conservation and accuracy. We present examples of state redistribution using two base schemes: MUSCL and a second order Method of Lines finite volume scheme. State redistribution is used to compute solutions to several standard test problems in gas dynamics on cut cell meshes, with both smooth and discontinuous solutions. We show that our method does not reduce the accuracy of the base scheme and that it successfully captures shocks in a non-oscillatory manner., Comment: 34 pages

A significant challenge in numerical solution to the Poisson-Boltzmann equation is due to singular charge sources in terms of Dirac delta functions. To overcome this difficulty, several regularization methods have been developed, in which the potential function is decomposed into two or three parts so that the singular component can be analytically solved using the Green's function, while other components become bounded. However, it was observed in the literature that some regularization methods are significantly less accurate than the others for unclear reasons, even though they are analytically equivalent. To understand this discrepancy, the numerical performance of four popular regularization methods is investigated in this work by implementing them with the Matched Interface and Boundary (MIB) approach, which is a sophisticated finite difference method for treating elliptic interface problems with discontinuous coefficients. With all four methods showing second order convergence, accuracy reduction is numerically observed in two schemes. This paper provides numerical analysis and experiment to trace the source of such reduction, and links the error to the fact that the Laplacian of Green's function is dropped outside the protein domain. While this term is analytically vanishing, its numerical negligence introduces a discretization error. Formulating via a proper elliptic interface problem, an effective accuracy recovery technique is proposed so that all four methods yield the same high precision. With this study, all involved regularization schemes are better understood and well connected into a unified framework.

According to the surface equivalence theorem [1] , any enclosed surface with electric and magnetic current densities enforced on it can generate exactly the same fields as the original problem. The problem is how to construct the current sources. In this paper, an efficient and accurate unified approach with an equivalent current density enforced on the outermost boundary is proposed to solve transverse magnetic (TM) scattering problems by objects embedded in multilayers. In the proposed approach, an equivalent current density is derived after the surface equivalence theorem is recursively applied on each boundary from innermost to outermost interfaces. Then, the objects are replaced by the background medium and the equivalent electric current density only on the outermost boundary is derived. The scattering problems by objects embedded in multilayers can be solved with the electric field integral equation (EFIE). Compared with the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) and other dual source formulations, the proposed approach shows significant benefits: only the surface electric current density instead of both the electric and magnetic current densities is required to model the complex objects in the equivalent problem. Furthermore, the electric current density is only enforced on the outermost boundary of objects. Therefore, the overall count of unknowns can be significantly reduced. At last, several numerical experiments are performed to validate its accuracy and efficiency. Although overhead is required to construct the intermediate matrices, the overall performance improvement is still significant as numerical results are shown. Therefore, it shows great potential useful in the practical engineering applications.

In this paper, we compute the band structure of one- and two-dimensional phononic composites using the extended finite element method (X-FEM) on structured higher-order (spectral) finite element meshes. On using partition-of-unity enrichment in finite element analysis, the X-FEM permits use of structured finite element meshes that do not conform to the geometry of holes and inclusions. This eliminates the need for remeshing in phononic shape optimization and topology optimization studies. In two dimensions, we adopt a rational Bezier representation of curved (circular) geometries, and construct suitable material enrichment functions to model two-phase composites. A Bloch-formulation of the elastodynamic phononic eigenproblem is adopted. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme—a method that uses Euler's homogeneous function theorem and Stokes's theorem to reduce integration to the boundary of the domain. Ghost penalty stabilization is used on finite elements that are cut by a hole. Band structure calculations on perforated (circular holes, elliptical holes, and holes defined as a level set) materials as well as on two-phase phononic crystals are presented that affirm the sound accuracy and optimal convergence of the method on structured, higher-order spectral finite element meshes. Several numerical examples are presented to demonstrate the advantages of p-refinement made possible by the spectral extended finite element method. In these examples, fourth-order spectral extended finite elements deliver O ( 10 − 8 ) accuracy in frequency calculations with more than thirty-fold fewer degrees-of-freedom when compared to quadratic finite elements.

In this paper, we reconsider the inverse Lax-Wendroff (ILW) procedure, which is a numerical boundary treatment for solving hyperbolic conservation laws, and propose a new approach to evaluate the values on the ghost points. The ILW procedure was firstly proposed to deal with the “cut cell” problems, when the physical boundary intersects with the Cartesian mesh in an arbitrary fashion. The key idea of the ILW procedure is repeatedly utilizing the partial differential equations (PDEs) and inflow boundary conditions to obtain the normal derivatives of each order on the boundary. A simplified ILW procedure was proposed in [28] and used the ILW procedure for the evaluation of the first order normal derivatives only. The main difference between the simplified ILW procedure and the proposed ILW procedure here is that we define the unknown u and the flux f ( u ) on the ghost points separately. One advantage of this treatment is that it allows the eigenvalues of the Jacobian f ′ ( u ) to be close to zero on the boundary, which may appear in many physical problems. We also propose a new weighted essentially non-oscillatory (WENO) type extrapolation at the outflow boundaries, whose idea comes from the multi-resolution WENO schemes in [32] . The WENO type extrapolation maintains high order accuracy if the solution is smooth near the boundary and it becomes a low order extrapolation automatically if a shock is close to the boundary. This WENO type extrapolation preserves the property of self-similarity, thus it is more preferable in computing the hyperbolic conservation laws. We provide extensive numerical examples to demonstrate that our method is stable, high order accurate and has good performance for various problems with different kinds of boundary conditions including the solid wall boundary condition, when the physical boundary is not aligned with the grids.

Partitioned solutions to fluid-structure interaction problems often employ a Dirichlet-Neumann decomposition, where the fluid equations are solved subject to Dirichlet boundary conditions on velocity from the structure, and the structure equations are solved subject to forces from the fluid. In some scenarios, such as an elastic balloon filling with air, an incompressible fluid domain may have pure Dirichlet boundary conditions, leading to two related issues which have been described as the incompressibility dilemma. First, the Dirichlet boundary conditions must satisfy the incompressibility constraint for a solution to exist. However, the structure solver is unaware of this constraint and may supply the fluid solver with incompatible velocities. Second, the constant fluid pressure mode lies in the null space of the fluid pressure solver, but must be determined to apply to the structure. Previously proposed solutions to the incompressibility dilemma have included modifying the fluid solver, the structure solver, or the Dirichlet-Neumann coupling interface between them. In this paper, we present a boundary pressure projection method which alleviates the incompatibility while maintaining the Dirichlet-Neumann structure of the decomposition and without modification of the fluid or solid solvers. Our method takes incompatible velocities from the structure solver and projects them to be compatible while in the process computing the constant pressure modes for the Dirichlet regions. The compatible velocities are then used as Dirichlet boundary conditions for the fluid while the constant pressure modes are added to the fluid-solver-computed pressures to be applied to the structure. The intermediate computation performed in the boundary pressure projection method is small, with the number of unknowns equal to the number of Dirichlet regions. We show that the boundary pressure projection method can be used to solve a variety of scenarios including inflation of an elastic balloon and the action of a hydraulic press. We also demonstrate the method on multiple coupled Dirichlet regions. The method offers a simple approach to overcome the incompressibility dilemma using a small intermediate computation that requires no additional knowledge of the black box fluid and solid solvers.

The aim of this paper is to study the feasibility of time-reversal methods in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine, from partial aperture boundary measurements, the presence and some physical properties of elastic unknown "inclusions", i.e. not observable solid objects, located in the elastic medium. We first derive a variational formulation of the acousto-elastic problem, from which one constructs a time-dependent finite element method to solve the forward, and then, the time reversed problem. Several criteria, derived from the reverse time migration framework, are then proposed to construct images of the inclusions, and to determine their locations. The dependence/sensitivity of the approach to several parameters (aperture, number of sources, etc.) is also investigated. In particular, it is shown that one can differentiate between a benign and malignant close inclusions. This technique is fairly insensitive to noise in the data., 22 pages, 15 figures

In this paper, we develop a high order finite difference boundary treatment method for the implicit-explicit (IMEX) Runge-Kutta (RK) schemes solving hyperbolic systems with possibly stiff source terms on a Cartesian mesh. The main challenge is how to obtain the solutions at ghost points resulting from the wide stencil of the interior high order scheme. We address this problem by combining the idea of using the RK schemes at the boundary and an inverse Lax-Wendroff (ILW) procedure in [29] , [30] . Specifically, we only apply the ILW procedure in the starting stage of the RK method. In the intermediate stages, we solve out the intermediate solutions as well as their first-order spatial derivatives at the boundary by using the RK schemes, which are then used to compute solutions at ghost points by Taylor expansions. Our method is different from the widely used approach for the explicit RK schemes by imposing boundary conditions at intermediate stages [14] , [29] , [30] which does not apply to IMEX schemes. In addition, the intermediate boundary conditions are available for explicit RK schemes only up to third order while our method applies to IMEX and explicit RK schemes of arbitrary order. The stability and the desired accuracy of our boundary treatment with IMEX RK method up to fourth order are demonstrated numerically through 1D examples and 2D reactive Euler equations.

This paper discusses the formulation of a class of hybrid compact schemes for structured irregular meshes. First, we derived three- and five-point compact schemes with required boundary closures for non-periodic problems. Second, we devised hybrid compact schemes from the explicit representation of the first- and second-order derivatives on irregular staggered and collocated meshes, respectively. For staggered meshes, hybrid interpolation schemes are also formulated. The main advantage of the hybrid formulation is that it requires boundary closures only for the compact scheme for first-order derivatives on the collocated mesh. Resolution and added numerical diffusion properties of hybrid schemes are assessed by using global spectral analysis (Sengupta et al., 2003). Direct numerical simulations of 2D dispersive shallow water and incompressible Navier-Stokes equations at different Reynolds numbers with fourth-order RK 4 time integration methods are considered to validate the accuracy and efficiency of new schemes. Numerical solutions are also compared with results reported in the literature.

Motivated by the need to handle complex boundary conditions efficiently and accurately in particle-in-cell (PIC) simulations, this paper presents a three-dimensional (3D) linear immersed finite element (IFE) method with non-homogeneous flux jump conditions for solving electrostatic field involving complex boundary conditions using structured meshes independent of the interface. This method treats an object boundary as part of the simulation domain and solves the electric field at the boundary as an interface problem. In order to resolve charging on a dielectric surface, a new 3D linear IFE basis function is designed for each interface element to capture the electric field jump on the interface. Numerical experiments are provided to demonstrate the optimal convergence rates in L 2 and H 1 norms of the IFE solution. This new IFE method is integrated into a PIC method for simulations involving charging of a complex dielectric surface in a plasma. A numerical study of plasma-surface interactions at the lunar terminator is presented to demonstrate the applicability of the new method.

A hybrid model for simulating rogue waves in random seas on a large temporal and spatial scale is proposed in this paper. It is formed by combining the derived fifth order Enhanced Nonlinear Schrödinger Equation based on Fourier transform, the Enhanced Spectral Boundary Integral (ESBI) method and its simplified version. The numerical techniques and algorithm for coupling three models on time scale are suggested. Using the algorithm, the switch between the three models during the computation is triggered automatically according to wave nonlinearities. Numerical tests are carried out and the results indicate that this hybrid model could simulate rogue waves both accurately and efficiently. In some cases discussed, the hybrid model is more than 10 times faster than just using the ESBI method, and it is also much faster than other methods reported in the literature.

E and Liu J. Comput. Phys. 138 (1997) 57-82 put forward a finite difference method for 3D viscous incompressible flows in the vorticity-vector potential formulation on non-staggered grids. In this paper, we will extend this method to the case of flows in the presence of a deformable surface. By use of two kinds of surface differential operators, the implementation of boundary conditions on a plane is generalized to a curved smooth surface with given velocity distribution, whether this be an inflow/outflow interface or a curved wall. To deal with the irregular and varying physical domain, time-dependent curvilinear coordinates are constructed and the corresponding tensor analysis is adopted in deriving the component form of the governing equations. Therefore, the equations can be discretized and solved in a regular and fixed parametric domain. Numerical results are presented for a 3D lid-driven cavity with a deforming surface and a 3D duct flow with a deforming boundary. A new way to validate numerical simulations is proposed based on an expression for the rate-of-strain tensor on a deformable surface.

In this paper we present a new algorithm for computing three-dimensional electrohydrodynamic flow in moving domains which can undergo topological changes. We consider a non-viscous, irrotational, perfect conducting fluid and introduce a way to model the electrically charged flow with an embedded potential approach. To numerically solve the resulting system, we combine a level set method to track both the free boundary and the surface velocity potential with a Nitsche finite element method for solving the Laplace equations. This results in an algorithmic framework that does not require body-conforming meshes, works in three dimensions, and seamlessly tracks topological change. Assembling this coupled system requires care: while convergence and stability properties of Nitsche's methods have been well studied for static problems, they have rarely been considered for moving domains or for obtaining the gradients of the solution on the embedded boundary. We therefore investigate the performance of the symmetric and non-symmetric Nitsche formulations, as well as two different stabilization techniques. The global algorithm and in particular the coupling between the Nitsche solver and the level set method are also analyzed in detail. Finally we present numerical results for several time-dependent problems, each one designed to achieve a specific objective: (a) The oscillation of a perturbed sphere, which is used for convergence studies and the examination of the Nitsche methods; (b) The break-up of a two lobe droplet with axial symmetry, which tests the capability of the algorithm to go past flow singularities such as topological changes and preservation of an axi-symmetric flow, and compares results to previous axi-symmetric calculations; (c) The electrohydrodynamical deformation of a thin film and subsequent jet ejection, which will account for the presence of electrical forces in a non-axi-symmetric geometry.

The article of record as published by be found at http://dx.doi.org/10.1016/j.jcp.2015.09.048 In computations, it is now common to surround artificial boundaries of a computational domain with a perfectly matched layer (PML) of finite thickness in order to prevent artificially reflected waves from contaminating a numerical simulation. Unfortunately, the PML does not give us an indication about appropriate boundary conditions needed to close the edges of the PML, or how those boundary conditions should be enforced in a numerical setting. Terminating the PML with an inappropriate boundary condition or an unstable numerical boundary procedure can lead to exponential growth in the PML which will eventually destroy the accuracy of a numerical simulation everywhere. In this paper, we analyze the stability and the well-posedness of boundary conditions terminating the PML for the elastic wave equation in first order form. First, we consider a vertical modal PML truncating a two space dimensional computational domain in the horizontal direction. We freeze all coefficients and consider a left half-plane problem with linear boundary conditions terminating the PML. The normal mode analysis is used to study the stability and well-posedness of the resulting initial boundary value problem (IBVP). The result is that any linear well-posed boundary condition yielding an energy estimate for the elastic wave equation, without the PML, will also lead to a well-posed IBVP for the PML. Second, we extend the analysis to the PML corner region where both a horizontal and vertical PML are simultaneously active. The challenge lies in constructing accurate and stable numerical approximations for the PML and the boundary conditions. Third, we develop a high order accurate finite difference approximation of the PML subject to the boundary conditions. To enable accurate and stable numerical boundary treatments for the PML we construct continuous energy estimates in the Laplace space for a one space dimensional problem and two space dimensional PML corner problem. We use summation-by-parts finite difference operators to approximate the spatial derivatives and impose boundary conditions weakly using penalties. In order to ensure numerical stability of the discrete PML, it is necessary to extend the numerical boundary procedure to the auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. Numerical experiments are presented corroborating the theoretical results. Moreover, in order to ensure longtime numerical stability, the boundary condition closing the PML, or its corresponding discrete implementation, must be dissipative. Furthermore, the numerical experiments demonstrate the stable and robust treatment of PML corners.

In this paper, a new Cartesian grid finite difference method is introduced based on the fourth order accurate matched interface and boundary (MIB) method and fast Fourier transform (FFT). The proposed augmented MIB method consists of two major parts for solving elliptic interface problems in two dimensions. For the interior part, in treating a smoothly curved interface, zeroth and first order jump conditions are enforced repeatedly by the MIB scheme to generate fictitious values near the interface. For the exterior part, two layers of zero-padding solutions are introduced beyond the original rectangular domain so that the FFT inversion becomes feasible. Different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations, can be imposed via the MIB scheme to generate fictitious values near boundaries. Based on fictitious values at both interfaces and boundaries, the augmented MIB method reconstructs Cartesian derivative jumps as auxiliary variables. Then, by treating such variables as unknowns, an enlarged linear system is obtained. In the Schur complement solution of such system, the FFT algorithm will not sense the solution discontinuities, so that the discrete Laplacian can be efficiently inverted. Therefore, the FFT-based augmented MIB not only achieves a fourth order of accuracy in dealing with interfaces and boundaries, but also produces an overall complexity of O ( n 2 log ⁡ n ) for a n × n uniform grid. Moreover, the augmented MIB scheme can provide fourth order accurate approximations to solution gradients and fluxes.

This paper is concerned with the boundary treatment of the lattice Boltzmann method (LBM) with general propagation for the incompressible Navier-Stokes equations. Since the propagation step involves three grid points, constructing effective boundary schemes for such method is highly nontrivial. To address this problem, we first construct an auxiliary boundary scheme under the diffusive scaling by using the Maxwell iteration. Based on this scheme and the interpolations of both distribution functions and fluid velocities, we propose a parameterized boundary scheme for the LBM with general propagation. The scheme thus obtained not only has second-order accuracy under the diffusive scaling for curved boundaries but also possesses good stability. Several specific schemes are also provided and their accuracy and stability are verified via four flow problems with both straight and curved boundaries. It should be noted that the proposed boundary scheme can also be used for the standard LBM.