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In this paper new innovative fourth order compact schemes for Robin and Neumann boundary conditions have been developed for boundary value problems of elliptic PDEs in two and three dimensions. Different from traditional finite difference operator approach, which may not work for flux type of boundary conditions, carefully designed undetermined coefficient methods are utilized in developing high order compact (HOC) schemes. The new methods not only can be utilized to design HOC schemes for flux type of boundary conditions but can also be applied to general elliptic PDEs including Poisson, Helmholtz, diffusion-advection, and anisotropic equations with linear boundary conditions. In the new developed HOC methods, the coefficient matrices are generally M-matrices, which guarantee the discrete maximum principle for well-posed problems, so the convergence of the HOC methods. The developed HOC methods are versatile and can cover most of high order compact schemes in the literature. The HOC methods for Robin boundary conditions and for anisotropic diffusion and advection equations with Robin or even Dirichlet boundary conditions are likely the first ones that have ever been developed. With the help of pseudo-inverse, or SVD solutions, we have also observed that the developed HOC methods usually have smaller error constants compared with traditional HOC methods when applicable. Non-trivial examples with large wave numbers and oscillatory solutions are presented to confirm the performance of the new HOC methods., Comment: 24 pages, 7 figures, 7 tables

Meta-biomaterials are applied to orthopedic implants to avoid stress shielding effects; however, there is no reason for the yield strength to be comparable to that of human bone. In this study, a composite unit cell was designed by combining the positive Poisson's ratio (PPR) and negative Poisson's ratio (NPR) unit cells, inspired by the second-phase strengthening theory. The purpose was to increase the strength while maintaining the elastic modulus. All structures were successfully fabricated from Ti-6Al-4V via selective laser melting. The relative density is between 0.08 and 0.24, which falls within the optimal range for bone growth. Mechanical tests indicated that the center of the inclined rod fractured in a stepwise fracture mode, which was consistent with the predictions of the Johnson-Cook model. The elastic modulus ranged from 0.652 ± 0.016 to 5.172 ± 0.021 GPa, and the yield strength varied from 10.62 ± 0.112 to 87.158 ± 2.215 MPa. An improved Gibson-Ashby law was proposed to facilitate the design of gradient structures. When the re-entrant angle was 40°, a hybrid body-centered cubic NPR structure was formed, resulting in a significant improvement in the mechanical properties. Importantly, the yield strength of the proposed composite structures increased by 43.23%, and the compression strength increased by 44.70% under the same elastic modulus. The strengthening mechanism has been proven to apply to other bending-dominated structures. Overall, this imparts unprecedented mechanical performance to auxetic meta-biomaterials and provides insights into improving the reported porous structures. STATEMENT OF SIGNIFICANCE: Auxetic meta-biomaterials exhibit auxetic properties that can improve the contact between the bone-implant interface and reduce the risk of aseptic failure. To avoid the stress shielding effect, the elastic modulus has traditionally been decreased by increasing the porosity. However, the strength is simultaneously reduced. Therefore, a composite unit cell was proposed to increase strength rather than modulus by combining the positive and negative Poisson's ratio unit cells, inspired by the second-phase strengthening theory. We observed a 43.23% increase in the yield strength of the composite structure without increasing the elastic modulus. This strengthening mechanism has been proven to apply to other bending-dominated structures. Our approach provides insights into improving other bending-dominated structures and broadening their applications for bone implantation.

SUMMARY The fast and accurate 3-D magnetotelluric (MT) forward modelling is core engine of the interpretation and inversion of MT data. In this study, we develop an improved extrapolation cascadic multigrid method (EXCMG) to solve the large sparse complex linear system arising from the finite-element (FE) discretization on non-uniform orthogonal grids of the Maxwell’s equations using potentials. First, the vector Helmholtz equation and the scalar auxiliary equation are derived from the Maxwell’s equations using Coulomb-gauged potentials. The weighted residual method is adopted to discretize the weak formulation and assemble the FE equation. Secondly, carefully choosing the preconditioned complex stable bi-conjugate gradient method (BiCGStab) as multigrid smoother, we develop an improved EXCMG method on non-uniform grids to solve the resulting large sparse complex non-Hermitian linear systems. Finally, several examples including three standard testing models (COMMEMI3D-1, COMMEMI3D-2 and DTM1.0) and a topographic model are used to validate the accuracy and efficiency of the proposed multigrid solver. Numerical results show that the proposed EXCMG algorithm greatly improves the efficiency of 3-D MT forward modelling, is more efficient than some existing solvers, such as Pardiso, incomplete LU factorization preconditioned biconjugate gradients stabilized method (ILU-BiCGStab) and flexible generalized minimum residual method with auxiliary space Maxwell preconditioner (FGMRES-AMS), and capable to simulate large-scale problems with more than 100 million unknowns.

Practical application of 3D magnetotelluric inversion requires efficient forward modeling of electromagnetic (EM) fields in the earth. To resolve realistic 3D structures, large computational domains and extremely large linear systems of equations are required. The iterative solvers, which are almost exclusively used to solve these systems, can be inefficient due to the abundant null space of the curl-curl operator. Multigrid (MG) solvers are considered a potentially efficient technique for solving such problems. However, due to the abundant null solution space and existence of the air layer, MG solvers can still converge slowly or even diverge. We have developed an efficient MG solver for finite-difference frequency-domain EM solution. In this algorithm, the excellent smoothing property of an efficient four-color cell-block Gauss-Seidel (GS) is exploited to remove the short-range errors effectively, and the interpolation and prolongation operators are used to handle the long-range errors. They work as a whole to speed the convergence of our algorithm remarkably. Because all of the nodes for the four-color cell-block GS are grouped into four colors and the edge components attached to different nodes in each color are completely decoupled, this can be used to develop a highly vectorized or parallelized algorithm. Another important property is that our algorithm is locally current divergence free, effectively eliminating spurious solutions in the null space of the curl-curl operator. The accuracy and efficiency of the algorithm are verified by comparing the numerical solutions obtained with our MG solver to those from the biconjugate gradient stabilized solver with different preconditioners based on synthetic models and a model from 3D inversion. Comparisons, in terms of iteration number and computational time, indicate that our algorithm is extremely stable and efficient relative to the other solvers. Our MG algorithm will be suitable for massively parallel computing as well.

We consider in this paper numerical approximations for the hydro-dynamically coupled phase-field model for an immiscible binary mixture of passive/active nematic Liquid Crystals (LCs) and viscous fluids. The passive model is a highly coupled nonlinear system that consists of the incompressible Navier-Stokes equations, the Cahn-Hilliard equations, and the constitutive equation for the polarization field. The active model can be viewed as an assembly of a passive part and a phenomenological active part. We then construct an efficient and linear time marching scheme to solve the passive model and further extend it to solve the active model. Various numerical simulations are presented to demonstrate the stability and accuracy of the proposed scheme in simulating some simulations in 2D and 3D, including the drop deformation dynamics and spinodal decompositions.

Magnetization is a natural property of magnetic materials which is widely used to study the ancient climate change whose footprint was recorded in rocks by palaeomagnetic data, uncover the location of unexploded ordnance, mineral deposits and gas and oil resources, and image regional crust and planet structure. An essential but very challenging task is to find analytical solutions of induced magnetic signals for general magnetic targets. In this study, for the first time, we demonstrate the existence of such solutions and explicitly give the analytical expressions of magnetic potential, magnetic field and magnetic gradient tensor for magnetic materials with polyhedral shapes and polynomial magnetization vectors. The order of magnetization polynomials can vary from linear order to arbitrary integer orders along both horizontal and vertical directions. Several synthetic and realistic iron deposit models are used to validate the accuracy and efficiency of our analytical solutions. Due to its inherent importance, our findings would have an immediate impact on magnetic explorations in earth science and have potentials in different applications.

SUMMARY The traditional gravity forward modelling methods for solving partial differential equations (PDEs) only can yield second-order accuracy. When computing the gravity field vector and gradient tensor from the obtained potential, those numerical differentiation approaches will inevitably lose accuracy. To mitigate this issue, we propose an efficient and accurate 3-D forward modelling algorithm based on a fourth-order compact difference scheme. First, a 19-point fourth-order compact difference scheme with general meshsizes in x-, y- and z-directions is adopted to discretize the governing 3-D Poisson’s equation. The resulting symmetric positive-definite linear systems are solved by the pre-conditioned conjugate gradient algorithm. To obtain the first-order (i.e. the gravity field vector) and second-order derivatives (i.e. the gravity gradient tensor) with fourth-order accuracy, we seek to solve a sequence of tridiagonal linear systems resulting from the above mentioned finite difference discretization by using fast Thomas algorithm. Finally, two synthetic models and a real topography relief are used to verify the accuracy of our method. Numerical results show that our method can yield a nearly fourth-order accurate approximation not only to the gravitational potential, but also to the gravity field vector and its gradient tensor, which clearly demonstrates its superiority over the traditional PDE-based methods.

In this paper, an efficient semi-implicit difference scheme for solving the fractional nonlinear Schrodinger equation with wave operator are proposed and analyzed. The semi-implicit scheme involves...

This paper is concerned with the numerical solutions of nonlinear Schrodinger equation (NLSE) in one-dimensional unbounded domain. The unbounded problem is truncated by the third-order absorbing boundary conditions (ABCs), and the corresponding initial-boundary value problem is solved by a three-level linearized difference scheme. By introducing auxiliary functions to simplify the difference scheme, the proposed difference scheme for NLSE with third-order ABCs is theoretically analyzed. It is strictly proved that the difference scheme is uniquely solvable and unconditionally stable, and has second-order accuracy both in time and space. Finally, several numerical examples are given to verify the theoretical results.

An extrapolation full multigrid (EXFMG) method is proposed for solving the large linear systems arising from linear finite element discretization of two- (2D) and three-dimensional (3D) elliptic bo...

Space and time approximations for two-dimensional space fractional complex Ginzburg–Landau equation are examined. The schemes under consideration are discreted by the second-order backward differential formula (BDF2) in time and two classes of the fractional centered finite difference methods in space. A linearized technique is employed by the extrapolation. We prove the unique solvability and stability for both numerical methods. The convergence of both numerical methods is analyzed at length utilizing the energy argument, and the convergence orders under the optimal step size ratio are O ( τ 2 + h 2 ) and O ( τ 2 + h 4 ) in the sense of the discrete L 2 -norm, where τ is the time step size, h = max { h x , h y } , and h x , h y are spatial grid sizes in the x -direction and y -direction, respectively. In addition, we construct a multistep alternating direction implicit (ADI) scheme and a multistep compact ADI scheme based on BDF2 for the efficiently numerical implementation. Finally, numerical examples are carried out to verify our theoretical results.

In this paper, we consider the numerical study for the multi-dimensional fractional-in-space Allen-Cahn equation with homogeneous Dirichlet boundary condition. By utilizing Strang's second-order splitting method, at each time step, the numerical scheme can be split into three sub-steps. The first and third sub-steps give the same ordinary differential equation, where the solutions can be obtained explicitly. While a multi-dimensional linear fractional diffusion equation needs to be solved in the second sub-step, and this is computed by the Crank-Nicolson scheme together with alternating directional implicit (ADI) method. Thus, instead of solving a multidimensional nonlinear problem directly, only a series of one-dimensional linear problems need to be solved, which greatly reduces the computational cost. A fourth-order quasi-compact difference scheme is adopted for the discretization of the space Riesz fractional derivative of α ( 1 α ≤ 2 ) . The proposed method is shown to be unconditionally stable in L 2 -norm, and satisfying the discrete maximum principle under some reasonable time step constraint. Finally, numerical experiments are given to verify our theoretical findings.

Krylov subspace solvers for frequency-domain electromagnetic forward modeling problems converge remarkably more slowly as the period increases. In this article, we present an efficient four-color cellblock Gauss Seidel (GS) preconditioner for finite-difference (FD) electromagnetic modeling in geophysical applications. Rather than updating the FD electromagnetic (EM) equation edge by edge, as in a traditional GS scheme, we renew six edge components attached to one node simultaneously (i.e., in cellblock manner) effectively enforcing a local divergence free condition for currents. To improve implementation efficiency, we reorder the nodes on the FD grid into four colors so that nodes in each color are uncoupled, allowing the use of highly parallel vectorized algorithms. The four-color cellblock GS preconditioner is implemented in the MATLAB code, in conjunction with a BiCGstab solver. It is compared, in terms of iteration number and computing time, with other three commonly used preconditioners [GS, symmetric successive overrelaxation (SSOR) and incomplete lower and upper triangular matrix decomposition (ILU)] on three models—two synthetic and one modified from the version of real data inversion. The comparison indicates that the proposed algorithm is extremely stable and efficient compared with the other three preconditioners tested, over a range of periods (1-1000 s). Especially at long periods, the improvement of our proposed algorithm is substantial. In addition, a parallel implementation of the cellblock GS preconditioner is straightforward due to the independence of nodes in each color.

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions, which is discretized by the space–time finite element method to propagate solutions. We develop a multigrid-reduction-in-time (MGRIT) algorithm with time-dependent time-grid propagators and provide its two-level convergence theory under the assumptions of the stability and simultaneous diagonalizability on time-grid propagators. Numerical results show that the proposed method possesses the saturation error order, theoretical results of the two-level variant deliver good predictions for our model problems, and significant speedups of the MGRIT can be achieved when compared to the two-level variant with F-relaxation (an equivalent version of the parareal algorithm) and the sequential time-stepping approach.

Finding the solution of the absolute value equations (AVEs) has attracted much attention in recent years. In this paper, we propose and analyze two generalized accelerated overrelaxation (AOR) methods for solving AVEs [Formula: see text], where [Formula: see text] is an [Formula: see text]-matrix. Furthermore, we discuss the convergence of the methods under some suitable assumptions. Numerical results are given to verify the effectiveness of our methods.

High order methods are preferred in many applications such as Helmholtz equations with large wave numbers to resolve the solution numerically. In this paper, a third order compact immersed interface method (IIM) based on the standard nine-point stencil is first proposed for solving Poisson/Helmholtz interface problems with discontinuous solutions and fluxes in two-space dimensions. Theoretically, new high order jump relations are derived, which are necessary for determining the correction terms of the finite difference scheme near or on an interface. Then, based on the developed third order compact IIM, an augmented third order compact finite difference method is further developed for elliptic interface problems with piecewise constant but discontinuous coefficients. In this approach, the jump in the normal derivative is set as an unknown so that the high order compact IIM can be applied. The co-dimension one augmented variable is solved by the Schur complement system via the GMRES iterative method. Various non-trivial examples are provided to show the performance of the new methods. One important feature of the new methods is that the computed normal derivative is also nearly third order accurate. Finally, the third order augmented method is applied to Poisson/Helmholtz equations on irregular domains with few changes along examples of Neumann, Robin, and Dirichlet boundary conditions.