Classical Finite Difference Scheme Research Articles

Explicit P1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell’s equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell’s equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space.

High order numerical methods for Vlasov-Poisson models of plasma sheaths

This article is a report of the CEMRACS 2022 project, called HIVLASHEA, standing for ”High order methods for Vlasov-Poisson models for sheaths”. A two-species Vlasov-Poisson model is described together with some numerical simulations, permitting to exhibit the formation of a plasma sheath. The numerical simulations are performed with two different methods: a first order classical finite difference (FD) scheme and a high order semi-Lagrangian (SL) scheme with Strang splitting; for the latter one, the implementation of (non-periodic) boundary conditions is discussed and different solvers for the electric field are compared. The codes are first evaluated on a one-species case, where an analytical solution is known. For the two-species case, cross-comparisons are performed and the numerical convergence of the SL method is investigated. The use of high order method is emphasized in this context. The (parallel) SL code turns out to be a robust tool to provide reliable numerical approximations of sheath stationary solution with realistic physical parameters.

An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ

<p>This work is the continuation of <sup>[<xref ref-type="bibr" rid="b11">11</xref>]</sup>, where a two-dimensional elliptic singularly perturbed weakly system, for which small parameters affected both the diffusion and the convection terms, was solved; moreover, all perturbation parameters could have different orders of magnitude, which is the most interesting and difficult case for this type of problem. It is well known that then, in general, overlapping regular or parabolic boundary layers appear in the solution of the continuous problem. To solve numerically the problem, the classical upwind finite difference scheme, defined on special piecewise uniform Shsihkin meshes, was used, proving its uniform convergence, with respect to all parameters, for four different ratios between them. In this paper, we complete the previous analysis, considering the two cases for these possible ratios, that were not considered in <sup>[<xref ref-type="bibr" rid="b11">11</xref>]</sup>. To see in practice the efficiency of the numerical method, we show the numerical results obtained with our algorithm for a test problem, when the cases analyzed in this work are fixed; from those results, the uniform convergence of the numerical algorithm follows, in agreement with the theoretical results.</p>

Three dimensional meshfree analysis for time-Caputo and space-Laplacian fractional diffusion equation

Numerical exploration of spatial fractional differential equations in three dimensions is not trivial due to their complexity, especially for complicated problem domains. In this work, the time-Caputo and space-Laplacian fractional diffusion equations in three dimensions are analyzed using a meshfree technique. In the temporal dimension, the classical L1 finite difference scheme is used to approach the Caputo fractional derivative. The spatial discretization is realized by a three dimensional reproducing kernel particle method (RKPM), which can eliminate the dependence of shape functions on certain meshes. Therefore RKPM is very suitable to approximate the field variable in complicated three dimensional domains compared with other mesh-dependent methods. For the purpose of increasing the computational efficiency, the stabilized conforming nodal integration (SCNI) and lumped mass matrix techniques are adopted in the Galerkin meshfree formulation. In the proposed method, the tedious derivatives computing for meshfree shape functions, numerical integration for Galerkin weak form and time-consuming inverse calculating for the large size mass matrix are all realized by more efficient approaches comparing with the conventional Galerkin RKPM. Several numerical examples in various domains with structured and unstructured discretization are studied to demonstrate the proposed methodology, and the results show very favorable performance of the proposed method regarding the accuracy and effectiveness.

Numerical solution of a malignant invasion model using some finite difference methods

Abstract In this article, one standard and four nonstandard finite difference methods are used to solve a cross-diffusion malignant invasion model. The model consists of a system of nonlinear coupled partial differential equations (PDEs) subject to specified initial and boundary conditions, and no exact solution is known for this problem. It is difficult to obtain theoretically the stability region of the classical finite difference scheme to solve the set of nonlinear coupled PDEs, this is one of the challenges of this class of method in this work. Three nonstandard methods abbreviated as NSFD1, NSFD2, and NSFD3 are considered from the study of Chapwanya et al., and these methods have been constructed by the use of a more general function replacing the denominator of the discrete derivative and nonlocal approximations of nonlocal terms. It is shown that NSFD1, which preserves positivity when used to solve classical reaction-diffusion equations, does not inherit this property when used for the cross-diffusion system of PDEs. NSFD2 and NSFD3 are obtained by appropriate modifications of NSFD1. NSFD2 is positivity-preserving when the functional relationship [ ψ ( h ) ] 2 = 2 ϕ ( k ) {\left[\psi \left(h)]}^{2}=2\phi \left(k) holds, while NSFD3 is unconditionally dynamically consistent with respect to positivity. First, we show that NSFD2 and NSFD3 are not consistent methods. Second, we tried to modify NSFD2 in order to make it consistent but we were not successful. Third, we extend NSFD3 so that it becomes consistent and still preserves positivity. We denote the extended version of NSFD3 as NSFD5. Finally, we compute the numerical rate of convergence in time for NSFD5 and show that it is close to the theoretical value. NSFD5 is consistent under certain conditions on the step sizes and is unconditionally positivity-preserving.

A robust uniformly convergent finite difference scheme for the time-fractional singularly perturbed convection-diffusion problem

The design and analysis of a finite difference method for solving a class of time-fractional singularly perturbed convection-diffusion problems is the subject of this paper. The time-fractional derivative is taken in the Caputo sense with the order α∈(0,1). The solution to this class of problems possesses weak singularity in the vicinity of the initial time t=0 and an exponential boundary layer at the right lateral surface as the singular perturbation parameter ε→0. To tackle the initial layer at time t=0, a classical L1 finite difference scheme is employed on a graded mesh to discretize the time-fractional derivative. Further, a standard upwinding procedure in the spatial direction is used on a piecewise uniform Shishkin mesh. It is shown that the resultant fully discrete scheme is parameter uniformly convergent with order of convergence O(M−p+N−1log⁡N), where p=min⁡{2−α,rα}, r is the graded mesh exponent and N,M are the number of mesh subintervals in space and time direction respectively. Lastly, numerical results illustrate that the proposed scheme is simple, efficient, robust and accurate as well as validate the error estimates presented in the paper.

Schwarzschild Harmonic Path Planning and Tetrahedral Mesh Smoothing

The Laplace––Beltrami operator is a fundamental tool for analysis on manifolds. It plays a vital role in differential geometry, geometry processing, data analysis and machine learning. Here we present a Laplace–Beltrami operator in spatial Schwarzschild manifolds, denoted by SLBO. Since spatial Schwarzschild manifold has global coordinates, we derive two global approximated schemes of the SLBO in 3-dimensional grids and tetrahedral meshes: (1) We discretize the SLBO by the seven-point stencil finite difference method. This derives a small correction of the classical finite difference scheme, and we prove the convergence for this scheme; (2) we give and prove a corrected-cotan formula of the SLBO. According to the two discrete schemes, the robot path planning and tetrahedral mesh smoothing, are designed to demonstrate the effectiveness of the proposed methods using the SLBO.

A novel numerical technique for solving time fractional nonlinear diffusion equations involving weak singularities

In this work, an efficient numerical approximation for the solution of the time fractional nonlinear diffuse interface model is studied. The solution to this problem has a weak singularity near the initial time . The fractional order nonlinear diffusion model is transformed into a system of nonlinear functional equations. The Daftardar–Gejji and Jafari method is employed to solve the corresponding nonlinear system. The L1 scheme is used to discretize the Caputo fractional derivative on a graded mesh in the time direction. In contrast, the spatial derivative is approximated by applying a classical central finite difference scheme to a uniform mesh. The convergence analysis and the error bounds are carried out. The analysis and the computational findings exhibit the effectiveness of the proposed method.

A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems

In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic singularly perturbed systems of convection-diffusion type, where the diffusion parameters can be different at each equation and even they can have different orders of magnitude. The numerical algorithm combines the classical upwind finite difference scheme to discretize in space and the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. We prove that if the spatial discretization is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. The technique used to discretize in time produces only tridiagonal linear systems to be solved at each time level; thus, from the computational cost point of view, the method we propose is more efficient than other numerical algorithms which have been used for these problems. Numerical results for several test problems are shown, which corroborate in practice both the uniform convergence and the efficiency of the algorithm.

Unconditionally positive NSFD and classical finite difference schemes for biofilm formation on medical implant using Allen-Cahn equation

Abstract The study of biofilm formation is becoming increasingly important. Microbes that produce biofilms have complicated impact on medical implants. In this paper, we construct an unconditionally positive non-standard finite difference scheme for a mathematical model of biofilm formation on a medical implant. The unknowns in many applications reflect values that cannot be negative, such as chemical component concentrations or population numbers. The model employed here uses the bistable Allen-Cahn partial differential equation, which is a generalization of Fisher’s equation. We study consistency and convergence of the scheme constructed. We compare the performance of our scheme with a classical finite difference scheme using four numerical experiments. The technique used in the construction of unconditionally positive method in this study can be applied to other areas of mathematical biology and sciences. The results here elaborate the benefits of the non-standard approximations over the classical approximations in practical applications.

Higher order robust numerical computation for singularly perturbed problem involving discontinuous convective and source term

In this article, we considered a singularly perturbed convection–diffusion equation having discontinuous convective and source terms. Due to these discontinuities, an expected behavior appears in the solution known as the interior layer behavior at the point of discontinuity. Layers are some small narrow regions of the domain where the steep variations in the solution occur as singular perturbation parameter ε approaches zero. Classical central finite difference schemes and finite element methods with piecewise polynomial basis functions on the uniform mesh are known to be inadequate to solve such problems unless an unacceptably large number of mesh points are used when ε is very small. Thus, the problem is solved numerically first by a first‐order accurate, simple upwind scheme on a specially designed piecewise uniform Shishkin mesh. Further, to enhance the order of convergence and to achieve better accuracy, we constructed a Richardson extrapolation scheme (RE scheme) whichis almost second‐order accurate. An extensive analysis has been carried out to establish the almost second‐order ε‐uniform convergence of the proposed RE scheme. Lastly, some numerical examples are considered to verify our theoretical results, and comparisons of numerical results are performed with the numerical results that exist in the literature.

Novel numerical techniques based on mimetic finite difference method for pricing two dimensional options

The Black–Scholes differential operator which underlies the option pricing of European and American options is known to be degenerate close to the boundary at zero. At this singularity, important properties of the differential operator are lost and the classical finite difference scheme applied to this problem fails to give accurate approximations as it is no longer monotone. In this paper novel numerical techniques based on mimetic finite difference method are proposed for accurately pricing European and American options. More precisely, we propose the mimetic and fitted mimetic finite difference methods, which are techniques that preserve and conserve general properties of the continuum operator in the discrete case. The fitted method further handles the degeneracy of the underlying partial differential equations (PDE). Those spatial discretization methods are coupled with the Euler implicit method for time discretization. Several numerical simulations are performed to demonstrate the robustness of our methods comparing to standard fitted finite volume method for both European and American put options.

A class of high-order finite difference schemes with minimized dispersion and adaptive dissipation for solving compressible flows

For the compressible flows with broadband length scales, good dispersion and dissipation properties are crucial for the numerical schemes to realize high-fidelity simulations. It has been recognized that the minimization of dispersion error is an effective measure to improve the resolution of schemes. However, it is still an open problem for the schemes to control or adjust the dissipation. In this paper, a class of high-order finite difference schemes with minimized dispersion and adaptive dissipation is proposed. As the first step to automatically adjust the dissipation according to the flow structures, we devise a scale sensor to quantify the local length scale of the numerical solution as the effective scaled wavenumber. Then the dispersion-dissipation condition is used to construct the relationship between the dissipation parameter and the effective scaled wavenumber. Thus, we obtain a class of finite difference schemes with adaptive dissipation. To achieve the shock-capturing capability and maintain the superior spectral properties in the smooth regions, we combine the adaptive dissipation scheme with the corresponding weighted essentially non-oscillatory (WENO) scheme to construct a hybrid scheme. At each grid point, either of the sub-schemes is applied dynamically according to a new shock detector. The shock detection accuracy of the new detector has been effectively improved compared with classical ones. The approximate dispersion relation (ADR) shows that the new hybrid scheme is accurate and robust. Several benchmark test problems with broadband length scales as well as discontinuities are presented to verify the high resolution, high efficiency and the good shock-capturing capability of the proposed scheme. The performance of the proposed scheme in turbulence simulation is also demonstrated by the three-dimensional isotropic Taylor-Green vortex problem.

An Analysis of the Robust Convergent Method for a Singularly Perturbed Linear System of Reaction–Diffusion Type Having Nonsmooth Data

In this paper, a coupled system of [Formula: see text] second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.

Numerical method for a boundary value problem for a linear singularly perturbed parabolic delay differential equation of reaction-diffusion type with discontinuous source term

A singularly perturbed boundary value problem for a linear parabolic second order delay differential equation of reaction-diffusion type with discontinuous source term is considered in the rectangle domain Ω = {(x, t) : 0 < x < 2,0 < t ⩽ T}. Discontinuity occurs in the source term at (d, t) < ∈ Ω. As the highest order space derivative is multiplied by a singular perturbation parameter, the components of the solution exhibit boundary layers at (x,t) = (0,t) and (x,t) = (2,t) and interior layers at (x, t) = (1, t) and/or at (x,t) = (d,t) and (x,t) = (1 + d, t) with respect to the position of (d, t) in Ω. A numerical method which uses classical finite difference scheme on a Shishkin piecewise uniform mesh is suggested to approximate the solution. The method is proved to be first order convergent uniformly for all the values of singular perturbation parameters. Numerical illustrations are presented so that the theoretical results are supported.

High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

In this paper, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the range of Knudsen number. Unfortunately, however, such schemes are not conservative. There are two main sources of lack of conservation. First, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points. However, for a small number of grids in velocity space such error is not negligible, because the parameters of the Maxwellian do not coincide with the discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme. As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number. To treat the first problem and ensure machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced by Mieussens. The second problem is treated by implementing a conservative correction procedure based on the flux difference form. In this way we can construct a conservative semi-Lagrangian scheme which is Asymptotic Preserving (AP) for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

A new stable finite difference scheme and its error analysis for two‐dimensional singularly perturbed convection–diffusion equations

AbstractThis work focuses on the numerical solution of two‐dimensional singularly perturbed convection–diffusion equations via a new stable finite difference (NSFD) scheme on a tensor product of two piecewise‐uniform Shishkin meshes. First, we convert the two‐dimensional equation into two one‐dimensional equations using the alternating direction implicit technique. A NSFD scheme has been developed using Taylor's series and the one‐dimensional equations. Here the truncation of Taylor's series is different from the classical finite difference scheme. The convergence analysis is also studied on a tensor product of two piecewise‐uniform Shishkin meshes. Numerical simulations confirm the theory. Moreover, from the numerical illustrations, it is also observed that the method is parameter uniform convergent.

Numerical Simulation of the Fractional Dispersion Advection Equations Based on the Lattice Boltzmann Model

The fractional dispersion advection equations (FDAEs) have recently attracted considerable attention due to their extensive application in the fields of science and engineering. For example, it has been shown that the anomalous solute transport behaviour that exists in hydrology can be well explained by introducing FDAEs. Therefore, the study of FDAEs has profound significance for understanding real transport phenomena in nature. Nevertheless, the existing algorithms for the FDAEs are generally intricate and costly. Therefore, exploiting an efficient solution technique has been a concern for scientists. In an effort to overcome this challenge, a promising lattice Boltzmann (LB) model for the FDAEs is presented in this paper. The Riemann–Liouville definition and the Grünwald–Letnikov definition are introduced for the time derivatives. In addition, Chapman–Enskog analysis is applied to recover the FDAEs. To test the validity of the model, three numerical examples are carried out. In addition, a comparative study of the proposed model and the classical implicit finite difference scheme is also conducted. The numerical results show that the model is suitable for simulating FDAEs.

A semi-discrete finite difference method to uniform stabilization of wave equation with local viscosity

In this paper, we propose a completely new and pure semi-finite difference scheme for uniform exponential convergence of approximation of 1-D wave equation with local viscosity damping by a semi-discrete finite difference scheme. It is known that for this partial differential equation system, the continuous system is exponentially stable yet the classical semi-discrete finite difference scheme is not uniformly exponentially stable. This paper adopts an order reduction finite difference scheme to semi-discretize the continuous system. There are at least three apparent advantages for this approach: a) it keeps the simplicity of the finite difference method; b) the uniformly exponentially convergence is preserved; c) the proof of the uniform exponentially stability is analogous to the continuous part. Finally, it is shown that the solution of the discrete system is convergent to the solution of the continuous counterpart and the energy of the discrete system is uniformly convergent to the continuous energy. The method is systematic to be applicable to any order of PDEs.

A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs

In this study, we consider the time-delayed singularly perturbed parabolic PDEs (SPPPDEs). We know that the classical finite difference scheme will not produce good results for singular perturbation problems on a uniform mesh. Here, we propose a new stable finite difference (NSFD) scheme, which produces good results on a uniform mesh and also on an adaptive mesh. The NSFD scheme is constructed based on the stability of the analytical solution. Results are compared with the results available in the literature and observed that the proposed method is efficient over the existing methods for solving SPPPDEs.