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In the current work, we present a variational mechanics framework for the coupled numerical prediction of cavitating turbulent flow and structural motion via a stabilized finite element formulation. To model the finite mass transfer rate in cavitation phenomena, we employ the homogenous mixture-based approach via phenomenological scalar transport differential equations given by the linear and nonlinear mass transfer functions. Stable linearizations of the finite mass transfer terms for the mass continuity equation and the reaction term of the scalar transport equations are derived for the robust and accurate implementation. The linearized matrices for the cavitation equation are imparted a positivity-preserving property to address numerical oscillations arising from high-density gradients typical of two-phase cavitating flows. The proposed formulation is strongly coupled in a partitioned manner with an incompressible 3D Navier-Stokes finite element solver, and the unsteady problem is advanced in time using a fully-implicit generalized-α time integration scheme. We first verify the implementation on the benchmark case of the Rayleigh bubble collapse. We demonstrate the accuracy and convergence of the cavitation solver by comparing the numerical solutions with the analytical solutions of the Rayleigh-Plesset equation for bubble dynamics. We find our solver to be robust for large time steps and the absence of spurious oscillations/spikes in the pressure field. The cavitating flow solver is coupled with a hybrid URANS-LES turbulence model. We validate the coupled solver for a very high Reynolds number turbulent cavitating flow over a NACA0012 hydrofoil section. Finally, the proposed method is applied in an Arbitrary Lagrangian-Eulerian framework to study turbulent cavitating flow over a pitching hydrofoil section and the characteristic features of cavitating flows such as re-entrant jet and periodic cavity shedding are discussed.

The moving particle semi-implicit method (MPS) is a well-known Lagrange method that offers advantageous in addressing complex fluid problems, but particle distribution is an area that requires refinement. For this study, a particle smoothing algorithm was developed and incorporated into the weakly compressible MPS (sWC-MPS). From the definition and derivation of basic MPS operators, uniform particle distribution is critical to numerical accuracy. Within the framework of sWC-MPS, numerical operators were modified by implementing coordinate transformation and smoothing algorithm. Modifying numerical operators significantly improved particle clustering, smoothed pressure distributions, and reduced pressure oscillations. To validate the numerical feasibility of the method, several cases were numerically simulated to compare sWC-MPS to the weakly compressible MPS (WC-MPS): a pre-defined two-dimensional (2-D) analytical function, Poiseuille's flow, Taylor Green vortex, and dam break. The results showed a reduction of errors caused by irregular particle distribution with lower particle clustering and smaller pressure oscillation. In addition, a larger Courant number, which represents a larger time step, was tested. The results showed that the new sWC-MPS algorithm achieves numerical accuracy even using a larger Courant number, indicating improved computational efficiency.

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem − ∇ 2 u = f in Ω ∖ Γ with Dirichlet boundary condition such that f is smooth in Ω ∖ Γ and the jump functions [ u ] and [ ∇ u ⋅ n → ] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of − ∇ 2 u = f + g δ Γ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇ u ⋅ n → are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω + and Ω − . The coefficient matrix A in the resulting linear system A x = b , following from the proposed scheme, is independent of any source term f, jump condition g δ Γ , interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω + or Ω − . The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

We present a variant of the immersed boundary method integrated with octree meshes for highly efficient and accurate Large-Eddy Simulations (LES) of flows around complex geometries. We demonstrate the scalability of the proposed method up to $\mathcal{O}(32K)$ processors. This is achieved by (a) rapid in-out tests; (b) adaptive quadrature for an accurate evaluation of forces; (c) tensorized evaluation during matrix assembly. We showcase this method on two non-trivial applications: accurately computing the drag coefficient of a sphere across Reynolds numbers $1-10^6$ encompassing the drag crisis regime; simulating flow features across a semi-truck for investigating the effect of platooning on efficiency., Comment: Accepted for publication at Computer and Mathematics with Applications

As a mesoscopic approach, the lattice Boltzmann method has achieved considerable success in simulating fluid flows and associated transport phenomena. The calculation, however, suffers from a massive amount of computing resources. A predictive model, to reduce the computing cost and accelerate the calculations, is proposed in this work. By employing an artificial neural network, composed of convolution layers and convolution long short-term memory layers, the model is an equivalent substitution of multiple time steps. A physical informed training loss function is introduced to improve the model predictive accuracy; and for the two-dimensional driven cavity problem, the mean square error of the prediction is less than 1.5 × 10 − 6 . For non-stationary flow, a time-dependent computing structure based on the current model is established. Nine iterative model calculations are performed consecutively for a two-dimensional driven cavity model, and the results are validated by comparing with the original (serial) lattice Boltzmann algorithm. Generally, in the case of training Reynolds number, for velocity and speed, the mean and the maximum absolute errors are lower than 0.012 and 0.12. Similarly, in the generalizing case, the mean and the maximum absolute errors are lower than 0.017 and 0.012. Besides, the current model's efficiency is about 15 times higher than that of the original lattice Boltzmann method.

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the M a c h number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.

In layered media, the solution of the Maxwell equations has discontinuity of the derivative or the function itself at media interfaces. For the first time, finite-difference schemes providing convergence for discontinuous solutions across straight media interfaces are proposed for the one-dimensional formulation of the Maxwell equations. These are bicompact conservative schemes. They are two-point and layer boundaries are taken as mesh nodes. The scheme explicitly accounts for physically correct interface conditions at media interfaces. We propose an essentially new technique which accounts for medium dispersion. All these features provide the second order of accuracy even on discontinuous solutions. Calculation examples, which illustrate these results, are given. The proposed method is verified by comparison with a previously performed experiment on propagation of normally incident plane wave on one-dimensional photonic crystal. Calculated spectrum agrees well with the measured one within experimental error of 2-5%.

In this work, we study at the L 2 – level global well-posedness as well as long-time stability of an initial-boundary value problem, posed on a bounded interval, for a generalized higher order nonlinear Schrodinger equation, modeling the propagation of pulses in optical fiber, with a localized damping term. In addition, we implement a precise and efficient code to study the energy decay of the higher order nonlinear Schrodinger equation and we prove its convergence and exponential stability of the discrete energy.

A novel near-wall modeling for large-eddy simulation based on the lattice Boltzmann method is proposed. An existing near-wall modeling based on the reconstruction of distribution functions is further developed to calculate high Reynolds number wall-bounded turbulent flow on a non-body-fitted Cartesian grid. The following three ideas are introduced into the existing reconstruction model: (1) the introduction of an image-point, (2) blending of sub-grid scale and Reynolds-averaged Navier-Stokes eddy viscosity considering the Reynolds shear stress resolved on a computational grid, and (3) modifications of velocity and eddy viscosity profiles near a wall. For the validation of the proposed model, numerical simulations of a turbulent channel flow on a non-body-fitted Cartesian grid are performed. Three different cases of the angles between the Cartesian grid line and the wall boundary are calculated, and the calculation results are compared with the results of direct numerical simulation or a semi-analytical profile in terms of streamwise velocity, resolved Reynolds shear, and normal stress profiles. The proposed model is robust and accurate, yielding highly satisfactory results regardless of the angles between the Cartesian grid line and the wall boundary.

With the objective of studying fluid-structure interaction (FSI) applications with sophisticated structural behavior, this work studies an approach that makes room for the use of complex structural models without penalizing the global computational time. The proposed approach is based on an algorithm that, simultaneously solving two subdomains, allows an asynchronous update of the boundary conditions at the interface according to the computational time spent by each of the solvers. The implementation of this algorithm revealed the appearance of numerical instabilities related to the parallel operation, and the independent evolution of the local solution in each subdomain. To overcome these stability problems, this work proposes a two-part stabilization strategy, both based on the use of a modal approach, as well as the adjustment of the formulation of the Jacobian of the fluid at the interface. The first part consists of a predictor-corrector method for the parallel operation and the second in an auxiliary coupling to assist the solution of the fluid domain. The impact of parameters such as the amount of correction, the number of modes and under-relaxation are studied. The use of the proposed asynchronous algorithm shows a positive impact when used for the solution of weakly coupled FSI applications (mass ratio m F / m S = 0.001 ). However, for applications with a larger or equivalent mass ratio, m F / m S = 0.1 or m F / m S = 1 , greater difficulty in stabilizing remain. Even though it is found that the proposed approach requires to be improved to ensure a robust solution, the encountered results represent an advance towards the use of an innovative algorithm in the FSI domain, one that could naturally offer an intermediary algorithm between the implicit external algorithm and the implicit internal algorithm.

Two efficient numerical schemes based on L 1 formula and Finite-Difference Time-Domain (FDTD) method are constructed for Maxwell's equations in a Cole-Cole dispersive medium. The temporal discretizations are built upon the leap-frog method and Crank-Nicolson method, respectively. We carry out the energy stability and error analysis rigorously by the energy method. Both schemes have been proved convergence with order O ( ( Δ t ) 2 − α + ( Δ x ) 2 + ( Δ y ) 2 ) , where Δ t , Δ x , Δ y are respectively the step sizes in time, space in x- and y-direction. The parameter α is a measure of the dispersion broadening. Numerical experiments are performed to confirm our theoretical analysis.

A high order discontinuous Galerkin method with Lagrange multiplier (DGLM) in space combined with discontinuous Galerkin (DG) method in time (DG-DGLM) [32] is numerically investigated for the approximation of the solution to the system of hyperbolic conservation laws. Computation is done in element by element fashion. P 0 time and space subcell limiting processes are applied to resolve the shocks. It is numerically shown that the high order DG-DGLM method is well-suited for long time integrations. Several numerical experiments for advection, shallow water, and compressible Euler equations are presented to show the performance of the high order DG-DGLM with P 0 time and space subcell limiting processes.

In this study, a linearized locally conservative scheme, based on using a weak Galerkin (WG)-mixed finite element method (MFEM), is developed for the Poisson–Nernst–Planck (PNP) system. In the dual-mixed formulation of the PNP equation, in addition to the three unknowns of concentrations p , n and the potential ψ , their fluxes, namely, σ p = ∇ p + p σ ψ and σ n = ∇ n − n σ ψ and σ ψ = ∇ ψ are introduced. These fluxes have an essential role in specifying the Debye layer and computing the electric current. The WG-MFEM considered here uses discontinuous functions to construct the approximation space. Also, a linearization scheme is employed to treat nonlinear terms. In the proposed method, the important physical laws of mass conservation and free energy dissipation are preserved without any restriction on the time step. Error estimates are developed and analyzed for both semi- and fully discrete WG-MFEM schemes. Furthermore, optimal error estimates (under adequate regularity assumptions on the solution) are derived. Several numerical results are provided and they demonstrate the efficiency of the proposed method and validate the convergence theorems.

We present in this work a unified framework for elliptic variational inequalities that gathers several problems in contact mechanics like the unilateral contact of one or two membranes or the Signorini problem. We study a family of Galerkin numerical schemes that discretize this framework. We prove the well-posedness of the discrete problem and we show that it is equivalent to a saddle-point mixed formulation containing complementarity constraints. To solve the arising nonlinear problem, we employ a semismooth Newton method and we prove local convergence properties. The abstract framework is then applied to the discretization of the unilateral contact between two membranes. We propose to discretize this problem with a finite element (FEM), a discontinuous Galerkin (dG), and a hybrid high-order (HHO) methods. We also adapt the semismooth Newton algorithm, including a static condensation procedure for the HHO method. Finally, we run numerical experiments for the FEM and HHO discretizations and compare their behavior.

Understanding the physics of droplet collisions is of crucial importance in many industrial processes such as inkjet printing, spray cooling, and spray combustion. In this paper, a modified multiple-relaxation-time pseudopotential multiphase lattice Boltzmann model is proposed to investigate binary droplet collision with different collision angles. The surface tension can be adjusted independently under the condition of high Reynolds number and large density ratio using this model. Then the model validation is performed by simulating three benchmark cases including stationary droplet, oscillating droplet, and capillary wave instability, respectively. Finally, different regimes of binary droplet collision are presented, and collision features such as liquid filament and satellite droplet are investigated at collision angles of 30°, 60°, 90°, 120°, and 150°. The results show that there is no liquid filament or satellite droplet at the collision angle of 30°, yet both of them appear as the angle increases. When the collision angle is 60°, the percentage of satellite droplet to liquid phase reaches 30%, while the proportions are less than 10% at collision angles of 90°, 120°, and 150°. Besides, two break-up mechanisms of “end-pinching” and capillary wave instability are inspected. It is found out that the back-flow phenomena at the joints of droplets and the liquid filament are caused by the “end-pinching” mechanism, which further leads to the capillary wave instability of the liquid filament.

In this paper we present a pseudospectral method in the disk. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions, as well as the biharmonic equation subject to nonhomogeneous Dirichlet conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary differential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this paper is applied to estimate Sherwood numbers integrating the mass flux to the disk, and it can be implemented to solve Lotka–Volterra systems and nonlinear diffusion problems involving chemical reactions.

This work is devoted to the development of a dissipative spatial discretization of a phase field model describing the dynamics of a multicomponent multiphase isothermal viscous compressible fluid with a resolved interface and effects associated with it (surface tension, wetting). Mass densities of mixture components are used as order parameters. The total Helmholtz free energy incorporates wall free energy providing wetting effects. The model under consideration is preliminary regularized, which allows one to increase time step in explicit time-marching methods. A finite-difference semi-discrete method is proposed and a discrete counterpart of the dissipativity theorem is proven. All the statements remain valid in the absence of regularization as well. 3D simulations of two-phase two-component and three-phase three-component fluids using explicit Euler method are performed. The simulation results confirm theoretical predictions.

In this paper, a novel system of Maxwell–Schrodinger equations with nonlocal effect in metamaterials is derived from the Drude model, hydrodynamical model and Schrodinger equation. A leap-frog finite element scheme, which can be solved one by one efficiently, is constructed by presenting a group of initial values. This scheme is proved to be stable conditionally in energy norm. It is confirmed that the error convergent rate is O ( τ 2 + h r ) by splitting the proof into three parts, where τ is the time step, h is the mesh size and r is the maximum total degree of polynomials in finite element spaces. Finally, some numerical results are given to verify the theories.

High dimensional conservative spatial distributed-order fractional diffusion equation is discretized by midpoint quadrature rule, Crank–Nicolson method, and a finite volume approximation, with alternating direction implicit scheme. The resulting scheme is shown to be consistent and unconditionally stable, hence convergent with order 3 − α , where α is the maximum of the involving fractional orders. Moreover, if the initial condition and source term possess Tensor-Train format (TT-format) with low TT-ranks, the scheme can be solved in TT-format, such that higher dimensional cases can be considered. Perturbation analysis ensures that the accumulated errors due to data recompression do not affect the overall convergence order. Numerical examples with low TT-ranks initial conditions and source terms, and with dimensions up to 20 are tested.

Splitter plates attached to a cylinder looking like hair (SPCH) is one of the self-adaptive devices used to control actual flow conditions, which in turn interact with Aeolian tones constituting a typical case of study in engineering industries. The direct numerical simulation of the sound waves stimulated by such devices is a complicated task due to the small levels of sound pressure and the time-consuming existing solvers. However, the Cumulant Lattice Boltzmann Method (LBM) provides stability and robustness at high Reynolds numbers and carries out these simulations satisfactorily. First, the fundamental acoustical properties of the Cumulant LBM are studied in this paper. Propagation of point and planar acoustic waves is considered including the temporal decay of a standing plane wave, the spatial decay of a planar acoustic pulse, and the propagation of spherical waves. Then, the Cumulant LBM as a fluid flow solver is coupled with a Finite Element structural mechanics solver to predict the effects of SPCH on the noise generated by cylinders at high Reynolds numbers as a practical fluid structure interaction (FSI) application. The spectral modification and possible acoustic damping impact of such flaps, plus the sound propagation from one and two circular cylinders are studied. A comparison of the theoretical and numerical results shows a reasonable capability of the Cumulant LBM to predict acoustical events with small errors in dissipation and dispersion. Furthermore, the results show that SPCH alter the phase of the vortex shedding cycle and decrease the transversal distance from the center line of the shed vortices. Flaps, thus, control the wake generated past a cylinder and have an effective impact on decreasing sound generation.

This paper describes the development of the Method of Regularized Sources for potential flow problems. It is based on the modification of the fundamental solution near the source point by replacing the singularity with a blob in form of a steep rational function. This allows to solve the problems in the same way as with Method of Fundamental Solutions, however without an artificial boundary. Method of Regularized Sources gives excellent results for Dirichlet boundary conditions, however it fails for Neumann boundary conditions. To overcome this problem are the source point positions on the segments of the boundary with Neumann boundary positions placed close to the collocation points. This approach somehow represents a blending of the Method of Regularized Sources and the classical Method of Fundamental Solutions. The novel approach is characterized by two free parameters; the blob thickness and the artificial boundary displacement position. A two-dimensional numerical example of potential flow around circle is analyzed in detail regarding these two free parameters. The modified Method of Regularized Sources gives even more accurate results for potential and derivatives than the Method of Fundamental Solutions. The source point can be placed 2–25 times closer to the boundary collocation points than with the classical Method of Fundamental Solutions and thus reduces the problem of the placement of the artificial boundary.

In this paper Double Multi Relaxation Time Lattice Boltzmann method (DMRT-LBM) is utilized to investigate the particle motion in a modeled room for the first time. The effect of indoor heating system location has been considered on both particle deposition and convective heat transfer so six different positions have been selected for the heating system. Results are shown for two Rayleigh numbers (104 and 105) and different particle sizes (1 μ m and 10 μ m). To study the particle motion, after the flow reaches steady state, 50 000 particles are randomly injected to the room and the number of deposited particles on the walls has been reported for 100 s. Considering the present results, when the heating system is in the middle of the wall, the maximum heat transfer rate is seen for both Rayleigh numbers. Also, the number of deposited 1 μ m particles on the walls for all locations of the heating system is zero for R a = 104, so the number of suspended particles is the maximum. For both Rayleigh numbers and 10 μ m particles, the number of deposited particles is maximum when the heating system is fixed on the top of the wall. It is also concluded that when the heating system is located on the top of the wall, the number of suspended 1 μ m particles for R a = 105 reaches its maximum value.

In this paper, we devise and analyse three highly efficient second-order accurate (in time) schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation where an asymmetric double-well potential function is considered. Based on the Scalar Auxiliary Variable (SAV) approach, we construct these schemes by splitting the FCH free energy in a novel and ingenious way. Utilizing the Crank–Nicolson formula, we firstly construct two semi-discrete second-order numerical schemes, which we denote by CN-SAV and CN-SAV-A, respectively. To be more specific, the CN-SAV scheme is constructed based on the fixed time step, while the CN-SAV-A scheme is a variable time step scheme. The BDF2-SAV scheme is another second-order scheme in which the fixed time step should be used. It is designed by applying the second-order backward difference (BDF2) formula. All the constructed schemes are proved to be unconditionally energy stable and uniquely solvable in theory. To the best of our knowledge, the CN-SAV-A scheme is the first unconditionally energy stable, second-order scheme with variable time steps for the FCH gradient flow equation. In addition, an effective adaptive time selection strategy introduced in Christlieb et al., (2014) is slightly modified and then adopted to select the time step for the CN-SAV-A scheme. Finally, several numerical experiments based on the Fourier pseudo-spectral method are carried out in two and three dimensions, respectively, to confirm the numerical accuracy and efficiency of the constructed schemes.

This article investigates the behavior of conjugate natural convection over a finite vertical surface immersed in a micropolar fluid in the presence of intense thermal radiation. The governing boundary layer equations are made dimensionless and then transformed into suitable form by introducing the non-similarity transformations. The reduced system of parabolic partial differential equations is integrated numerically along the vertical plate by using an implicit finite difference Keller-box method. The features of fluid flow and heat transfer characteristics for various values of micropolar or material parameter, K , conjugate parameter, B , and thermal radiation parameter, R d , are analyzed and presented graphically. Results are presented for the local skin friction coefficient, heat transfer rate and couple stress coefficient for high Prandtl number. It is found that skin friction coefficient and couple stress coefficient reduces whereas heat transfer rate enhances when the micro-inertia parameter increases. All the physical quantities get augmented with thermal radiation.

Due to the complexity of detailed nuclear reactor numerical simulation, especially the complicated geometry and multi-physics coupling properties of the advanced reactor, a nuclear reactor kinetics solver for nuclear reactor engineering and design needs to be developed. Based on the open source C++ software OpenFOAM, this work establishes a neutron transport kinetics solver, namely ntkFoam, for the nuclear reactor kinetics simulation, from the governing equations to the detailed implementations. The coupling between multi-group neutron transport equations and the delayed neutron precursor balance equations are considered with using the finite volume method. By introducing the inverse power method to the OpenFOAM, the k-eigenvalue problems are calculated, and by coupling the neutron transport and delayed neutron precursor calculations using the Euler implicit scheme, the transient kinetics problems are simulated. Numerical results show that the proposed ntkFoam can simulate the multi-group neutron transport kinetics problem accurately and flexibility, and both the regular and irregular mesh configurations can be adopted. This work can provide some new perspectives and foundations to nuclear reactor coupling calculations.

The imposition of inhomogeneous Dirichlet (essential) boundary conditions is a fundamental challenge in the application of Galerkin-type methods based on non-interpolatory functions, i.e., functions which do not possess the Kronecker delta property. Such functions typically are used in various meshfree methods, as well as methods based on the isogeometric paradigm. The present paper analyses a model problem consisting of the Poisson equation subject to non-standard boundary conditions. Namely, instead of classical boundary conditions, the model problem involves Dirichlet- and Neumann-type nonlocal boundary conditions. Variational formulations with strongly and weakly imposed inhomogeneous Dirichlet-type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non-uniform rational B-splines (NURBS). The attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods.

A distributed elliptic control problem with control constraints is considered, which is formulated as a three field problem and consists of two variational equations for the state and the co-state variables as well as of a variational inequality for the control variable. Two discretization approaches with h p -finite elements are discussed: In the first discretization approach all variables (state, co-state and control) are discretized. A semi smooth Newton method is introduced for solving the resulting algebraic system. In the second discretization approach only the state and the co-state variables are discretized, whereas the control is determined by projection. A simple fixed point scheme is presented for the iterative solution of this approach. The main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates, which enables h p -adaptive mesh refinements. In particular, the estimates can be applied to the iteration solutions, so that they can be used as a stopping criterion of the iterative solution schemes. In several numerical experiments the order of convergence of the (adaptive) discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied.

In this paper, we propose a linear, unconditional energy stable time discretization scheme for Cahn–Hilliard–Navier–Stokes model, which is a phase-field model for two-phase incompressible flow. Based on a Lagrange multiplier approach, our proposed scheme is linearized by using implicit–explicit treatments, and the computation of the velocity field u , the pressure p , the phase function ϕ are decoupled. Using the mathematical induction, the bound ‖ ϕ n ‖ L ∞ is obtained. Rigorous error analysis for the variable u , p and ϕ are carried out in semi-discrete form. Finally, some numerical experiments are done to demonstrate the effectiveness for the proposed method.

In this paper we propose a new distribution of Gaussian points to compute the weak form integrals of the Generalized Finite Element Method (GFEM). The relevance of this new distribution is the possibility of evaluating the integrals of oscillatory functions inside the reference triangle in an alternate way. A simple scheme of relocation of the quadrature points allows to improve the efficacy of the method. A wave propagation problem is solved with the proposed technique and its performance is compared to conventional and other existing proposals. In addition, we propose a new distribution of Gaussian points to the tetrahedral reference element

In this paper, we have proposed a higher order non-linear PDE model containing diffusion terms of T V − L 2 and T V − H − 1 model for effective image denoising. Fourier spectral method for space and convexity splitting method for time have been used to solve the proposed PDE model. Stability analysis for the time discretized equation has been derived. Numerical experiment has been done on some test images and the results have been compared with the results of some existing models.

Relaxation models for the pressureless gas dynamics (PGD) equations attempt to satisfy the strictly hyperbolic conservation law in order to employ the well-posed approximated Riemann solvers. In this study, a new type of relaxation model is proposed to resolve two shortcomings of the existing relaxation models: the constant propagation speed of sound, and the collapse of delta shock waves in multidimensional problems. The proposed model seeks a strictly hyperbolic system of equations without any special consideration for the proper values of the propagation speed of sound. Numerical tests showed that the proposed model can accurately describe the behavior of the PGD equations, in particular, the occurrence of delta shock waves and vacuum states in a multidimensional problem.

In the present work, a generalized finite difference method (GFDM), a meshless method based on Taylor-series approximations, is proposed to solve stationary 2D and 3D Stokes equations. To overcome the troublesome pressure oscillation in the Stokes problem, a new simple formulation of boundary condition for the Stokes problem is proposed. This numerical approach only adds a mixed boundary condition, the projections of the momentum equation on the boundary outward normal vector, to the Stokes equations, without any other change to the governing equations. The proposed formulation can be easily discretized by the GFDM. The GFDM is evolved from the Taylor series expansions and moving-least squares approximation, and the derivative expressed of unknown variables as linear combinations of function values of neighboring nodes. Numerical examples are utilized to verify the feasibility of the proposed GFDM scheme not only for the Stokes problem, but also for more involved and general problems, such as the Poiseuille flow, the Couette flow and the Navier–Stokes equations in low-Reynolds-number regime. Moreover, numerical results and comparisons show that using the GFDM to solve the proposed formulation of the Stokes equations is more accurate than the classical formulation of the pressure Poisson equation.

The present work compares fluid–solid interface approaches for the lattice Boltzmann method (LBM) to study vortex-induced vibrations (VIV). Two classes of fluid–solid interface approaches, namely the partially saturated methods (PSM) (Noble and Torczynski (1998), Holdych (2003), Krause et al. (2017), Trunk et al. (2018)) and moving boundary methods (MBM) (Bouzidi et al. (2001), Lallemand and Luo (2003), Yu et al. (2003), Ginzburg et al. (2008) and Filippova and Hanel (1998), Mei et al. (1999, 2000)), are investigated. First, the Galilean invariance of aerodynamic coefficients obtained by each scheme is examined. The Bhatnagar–Gross–Krook (BGK) LBM is utilized to simulate an eccentrically positioned cylinder in a transient Couette flow. In addition, various refill methods for MBM and volume approximation techniques for PSM are tested. Besides different error norms and a grid independence study of each method, the Galilean invariance violating frequencies are studied as well. These error calculations are used to choose a representative for each class to simulate vortex shedding. The VIV test case describes a transverse oscillation of a cylinder in a freestream at a Reynolds number of 100. Both free and forced cylinder oscillations are examined to study known phenomena as lock-in and lock-out zones. The results of MBM and PSM are in good agreement to literature values and prove the suitability for VIV simulations.

We consider an inverse problem where the forward problem is that of linear plane stress elasticity, or equivalently, that of linear heat/hydraulic conduction. We demonstrate that the linearized version of the saddle point problem obtained from the minimization problem inherits some stability from the forward elliptic problem. In particular, it is stable for the response variable and the Lagrange multiplier, but not for the material property field. This lack of stability implies that we are unable to prove optimal convergence with mesh refinement for the overall problem. We overcome this difficulty by adding to the saddle point problem a residual-based term that provides sufficient stability, and prove optimal convergence in an energy-like norm. We verify these estimates through simple numerical examples. We note that while we have considered a specific model for an inverse elliptic problem in this manuscript, similar ideas could be developed for a broad class of inverse elliptic problems.

This paper introduces a novel method for deducing high-order finite difference schemes for the three-dimensional Helmholtz equation, Δ p ± λ 2 p = f . The new method can reach an eighth-order of accuracy. It is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives. The scheme is obtained from Taylor series expansion and plane wave theory analysis, and it is constructed from a few modifications to the standard centered finite differences. A classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given. Numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of λ -values. Moreover, the numerical method is capable to solve with great precision the monotone and oscillatory Helmholtz problem with variable λ -values. Thus, the proposed formulation results in an attractive method, easy to implement, with high order of accuracy but nearly the same computational cost as those of an explicit formulation.

In this paper, an efficient linear compact alternating direction implicit (CADI) scheme with second order temporal accuracy and fourth order spatial accuracy is proposed for approximating solution to the initial–boundary problem for the two-dimensional Klein–Gordon–Schrodinger equations. Furthermore, maximum norm error estimates of numerical solutions are obtained by using the energy method and the mathematical induction method. Finally, some numerical experiments are presented to support the theoretical results.

A simple direct heating thermal immersed boundary-lattice Boltzmann method for buoyancy-driven incompressible flow with complex geometry boundaries is developed to simulate natural convection with curved boundary. In the newly developed method, both Dirichlet and Neumann boundary conditions in macroscopic equation are deduced into a novel mesoscopic discrete heat source. The mesoscopic discrete heat source consists of simplified explicit discrete heat source scheme which reduces the computational complexities and is introduced into lattice Boltzmann equation for temperature field, which extends the idea of the previous direct forcing IB-LBM. Both isothermal boundary condition (i.e. Dirichlet boundary condition) and constant heat flux boundary condition (i.e. Neumann boundary condition) are considered in our simulations. The efficiency and accuracy of the present method are demonstrated by simulating both two dimensional and three dimensional thermal flow with curved boundaries. Numerical results indicate that the efficiency and accuracy of the present method are well consistent with experimental and the other numerical results. The present method has also been successfully applied to three dimensional simulations of natural convection in a thin annulus with adiabatic wall at both vertical sides.

Recently, relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation. Here, we generalize this approach to guarantee local entropy inequalities for finitely many convex functionals (entropies) and apply the resulting methods to the compressible Euler and Navier–Stokes equations. Based on the unstructured h p -adaptive SSDC framework of entropy conservative or dissipative semidiscretizations using summation-by-parts and simultaneous-approximation-term operators, we develop the first discretizations for compressible computational fluid dynamics that are primary conservative, locally entropy stable in the fully discrete sense under a usual CFL condition, explicit except for the parallelizable solution of a single scalar equation per element, and arbitrarily high-order accurate in space and time. We demonstrate the accuracy and the robustness of the fully-discrete explicit locally entropy-stable solver for a set of test cases of increasing complexity.

Under-resolved direct numerical simulations of turbulence in the wake of a single-square-grid (SSG) are conducted using the lattice-Boltzmann method (LBM), with varying degrees of partial moment entropic stabilisation. Turbulence statistics along the grid centreline show satisfactory agreement with literature, greatly outperforming a control study employing the single-relaxation-time Smagorinsky LES model with equivalent resolution. The coarse-resolution entropic LBM simulations are able to capture the qualitative behaviour of the anisotropic and non-homogeneous​ turbulence generated by the SSG, indicating possible extension of the capability of the entropic lattice-Boltzmann method to simulate more exotic forms of turbulence beyond those reported by the majority of current studies. However, in the extreme near field of the single-square-grid, pseudo-compressibility effects, even at a low lattice Mach number, undermine LBM predictions for turbulence which is generated by action of the pressure Hessian — this is found to be an issue for all models studied, and not a numerical artefact of the entropic stabilisation. With regard to the action of the entropic stabiliser, a comparison of models employing differing degrees of stabilisation on higher-order natural moments suggest that the trace of the stress tensor should not be subject to entropic control for better accuracy in turbulence prediction.

The cable equation plays a significant role in many areas of electrophysiology and in modeling neuronal dynamics. In recent years, considerable attention has been devoted to distributed-order differential equations because they appear to be more effective for modeling complex processes. In this work, a finite difference/Legendre spectral method is presented for the numerical simulation of the two-dimensional (2D) distributed-order time-fractional cable equation, where the finite difference method is employed in the temporal discretization and Legendre spectral method is adopted in the spatial discretization. The midpoint quadrature rule is used to approximate the distributed-order, such that the considered equation could be transformed into a multi-term fractional equation. The stability and convergence analysis of the proposed scheme is established, which illustrates that the numerical solution converges to the exact solution with order O ( τ 2 + σ 2 + N − s ) , where τ , σ , N are the time step size, the step length in the approximation of the distributed-order and the polynomial degree, respectively. Furthermore, to demonstrate the versatility and applicability of our method, we provide numerical results that show good agreement with the theoretical analysis.

This research paper introduces a robust, efficient and stabilized midpoint quadrature to generate element stiffness matrices for quadrilateral and hexahedral elements in finite element method (FEM). This new quadrature is established based on Passive Richardson extrapolation which has the inherent ability to stabilize and improve the accuracy of the extrapolated results. The new quadrature includes a simple artificial stabilizing function with Gauss one-point quadrature (1GP) as an anti-hourglass effect. For the first approximation, the 1GP (sampling point to be considered at origin (0,0)) is done to evaluate element matrices (thermal conductivity matrices, capacitance matrices etc.,) in the 4-square element ( [-2, 2] x [-2, 2] ) in the ( ξ , η ) plane. As a stabilizing function, sampling points are considered to be either at the center of the four element edges or center of the four quadrants in the reference square element with appropriate constant weighting functions. For hexahedral elements also, the element stiffness coefficients are calculated only in the center of the element and center of all the octants of the reference cubic element ([-2, 2] x [-2, 2] x [-2, 2]), the proposed method deals mostly with integer variables and sampling at (0,0,0), whereas Gauss quadrature uses lengthy float variables corresponding to irrational numbers in terms of decimals in the stiffness matrix calculations, thus consuming time. A number of standard example problems are demonstrated to assess the efficiency of the proposed method. The results of all the numerical examples are found to be in good agreement with the exact results and conventional Gauss quadrature results.

We design and analyze a new finite element discretization technique based on iterative two-grid methods for semilinear elliptic partial differential equations (PDEs). We first present an iterative two-grid method that just applies the classical two-grid for semilinear elliptic PDEs in a successive fashion, namely, to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space. Thus, we only need to deal with semilinear problems in the coarse space in the first algorithm. Secondly, we replace the semilinear term by using the corresponding first order Taylor expansion in the second iterative two-grid. We provide the error estimates for the second iterative algorithm and present numerical experiments to show the efficiency of our methods.

This paper deals with the numerical computation and analysis for a class of two-dimensional time–space fractional convection–diffusion equations. An implicit difference scheme is derived for solving this class of equations. It is proved under some suitable conditions that the derived difference scheme is stable and convergent. Moreover, the convergence orders of the scheme in time and space are also given. In order to accelerate the convergence rate, by combining the Kronecker product splitting (KPS) preconditioner with the generalized minimal residual (GMRES) method, a preconditioning strategy for implementing the difference scheme is introduced. Finally, several numerical examples are presented to illustrate the computational accuracy and efficiency of the methods.

Most high-order methods often suffer from two- and three-dimensional unstructured grids, particularly, for supersonic and hypersonic gas flows. We applied a high-order mixed-modal discontinuous Galerkin method with a coupled slope limiter to investigate multidimensional cases of supersonic and hypersonic gas flows. Furthermore, we provide detailed implementations and validations of two- and three-dimensional problems. Examples include double Mach reflection problems and supersonic and hypersonic gas flows around a cylinder. In addition, we provide case studies of a 3D hypersonic vehicle to evaluate its computational performance on complex hypersonic gas flows. This study can be regarded as an attempt to apply the discontinuous Galerkin method to engineering problems.

In this paper, we mainly propose some new preconditioners for a special class of 3 × 3 block saddle point problems, which can arise from the time-dependent Maxwell equations and some other practical problems. Firstly, the solvability of this kind of problem is investigated under suitable assumptions. Then, we analyze the corresponding eigenvalues of the new preconditioned matrices presented in this work. Furthermore, we show that proposed preconditioned matrices only have at most three distinct eigenvalues. Finally, two numerical examples are also carried to verify the effectiveness of proposed preconditioners.

The problem of reconstructing the initial condition in the Neumann problem for linear parabolic equations with space-and-time-dependent coefficients from noisy observation of the solution on a part of the boundary is studied. Three variational methods are suggested for solving the problem: 1) the least squares method which aims at the minimizing the misfit between the observation on the boundary by varying the initial condition, 2) J.-L. Lions’ method (proposed in 1968) which minimizes the gap between the solutions to the corresponding Neumann and Dirichlet problems when the observation is taken on whole boundary, and 3) energy space approach which minimizes an energy-like functional measuring the gap between the solutions to the corresponding Neumann and Dirichlet problems when the observation is taken on whole boundary. For these problems, we provide the gradient of the functionals to be minimized and derive the first optimality conditions. To solve the problems numerically, we discretize the problems either by the finite element method or by the boundary element method. The error estimates are proved and numerical examples are tested which show the efficiency of our approaches.

Based on two-grid algorithm, we develop the superconvergence analysis of fully-discrete scheme with the lowest-order Nedelec element and Crank–Nicolson scheme for the magneto-heat coupling model, which is also considered as Maxwell’s equations with a thermal effect. Then, main process of numerical analysis has two parts: On one hand, we utilize Newton-type Taylor expansion on superconvergent solutions for those nonlinear terms on coarse mesh, differing from the numerical solution, which makes our given two-grid method successful. On the other hand, we prove the solutions on the fine mesh to be higher accuracy by the postprocessing interpolation technique. Such a design is conducive to improving the computational accuracy in space and decreasing time consumption simultaneously. By employing the skill, we can obtain the convergent rate of O ( Δ t 2 + h 2 + H 3 ) , which means that the space mesh size should satisfy h = O ( H 3 2 ) . Finally, the numerical example is presented to verify the theoretical results.

A stabilized interface method is presented for monolithic coupling of thermomechanical fields across incompatible internal interfaces. Embedding Discontinuous Galerkin (DG) ideas in the Variational Multiscale (VMS) framework yields interface coupling terms that accommodate nonmatching discretizations across glued meshes. A unique feature of the method that emanates from fine-scale modeling facilitated by VMS is the derivation of analytical expressions for the Lagrange multipliers that enforce continuity of the fields across nonmatching meshes. This stabilized form is free of any user-defined parameters and renders itself to consistent linearization that leads to quadratic convergence of the method. The coupling terms at the interface also provide an avenue to employ various kinematic models for interfacial response in multi-constituent materials. An interfacial gap/crack function is introduced together with the conjugate traction vector to model discrete interfacial failure in multi-material applications. An algorithm for consistent implementation of interfacial models for tension, compression, and sliding friction across non-conforming meshes is presented. Test cases that employ 8-node hexahedra and 4-node tetrahedra are presented that highlight the generality of the method and validate its robustness for mathematically nonsmooth thermomechanical problems.

International audience; In the present contribution, we consider Helmholtz equation with material coefficients being constants in each subdomain of a geometric partition of the propagation medium (discarding the presence of junctions), and we are interested in the numerical solution of such a problem by means of local multi-trace boundary integral formulations (local-MTF). For a one dimensional problem and configurations with two subdomains, it has been recently established that applying a Jacobi iterative solver to local-MTF is exactly equivalent to an Optimised Schwarz Method (OSM) with a non-local impendance. In the present contribution, we show that this correspondance still holds in the case where the subdomain partition involves an arbitrary number of subdomains. From this, we deduce that the depth of the adjacency graph of the subdomain partition plays a critical role in the convergence of linear solvers applied to local-MTF: we prove it for the case of homogeneous propagation medium and show, through numerical evidences, that this conclusion still holds for heterogeneous media. Our study also shows that, considering variants of local-MTF involving a relaxation parameter, there is a fixed value of this relaxation parameter that systematically leads to optimal speed of convergence for linear solvers.

We study the backward problem of determining the initial condition for a system of parabolic diffusion equations, which is severely ill-posed in the sense of Hadamard. To stabilize the solution, we develop the quasi-reversibility (QR) and Fourier truncation methods to construct the regularized solutions. We also investigate the error estimates and convergence rates between the regularized solutions and the true solution in L 2 − and H 1 − norm. A numerical scheme is presented.