Higher-order accurate two-step finite difference schemes for the many-dimensional wave equation

High-order accurate entropy stable adaptive moving mesh finite difference schemes for special relativistic (magneto)hydrodynamics

This paper develops high-order accurate well-balanced (WB) energy stable (ES) finite difference schemes for multi-layer (the number of layers M⩾2) shallow water equations (SWEs) with non-flat bottom topography on both fixed and adaptive moving meshes, extending our previous work on the single-layer shallow water magnetohydrodynamics [25] and single-layer SWEs on adaptive moving meshes [58]. To obtain an energy inequality, the convexity of an energy function for an arbitrary M is proved by finding recurrence relations of the leading principal minors or the quadratic forms of the Hessian matrix of the energy function with respect to the conservative variables, which is more involved than the single-layer case due to the coupling between the layers in the energy function. An important ingredient in developing high-order semi-discrete ES schemes is the construction of a two-point energy conservative (EC) numerical flux. In pursuit of the WB property, a sufficient condition for such EC fluxes is given with compatible discretizations of the source terms similar to the single-layer case. It can be decoupled into M identities individually for each layer, making it convenient to construct a two-point EC flux for the multi-layer system. To suppress possible oscillations near discontinuities, WENO-based dissipation terms are added to the high-order WB EC fluxes, which gives semi-discrete high-order WB ES schemes. Fully-discrete schemes are obtained by employing high-order explicit strong stability preserving Runge-Kutta methods and proved to preserve the lake at rest. The schemes are further extended to moving meshes based on a modified energy function for a reformulated system, relying on the techniques proposed in [58]. Numerical experiments are conducted for some two- and three-layer cases to validate the high-order accuracy, WB and ES properties, and high efficiency of the schemes, with a suitable amount of dissipation chosen by estimating the maximal wave speed due to the lack of an analytical expression for the eigenstructure of the multi-layer system.

Numerical analysis of a high-order accurate compact finite difference scheme for the SRLW equation

Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers' equation

A reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation

One-dimensional heat equation was solved for different higher-order finite difference schemes, namely, forward time and fourth-order centered space explicit method, backward time and fourth-order centered space implicit method, and fourth-order implicit Crank-Nicolson finite difference method. Higher-order schemes have complexity in computing values at the neighboring points to the boundaries. It is required there a specification of the values of field variables at some points exterior to the domain. The complexity was incorporated using Hicks approximation. The convergence and stability analysis was also computed for those higher-order finite difference explicit and implicit methods in case of solving a one dimensional heat equation. The obtained numerical results were compared with exact solutions. It is found that backward time and fourth-order centered space implicit scheme along with Hicks approximation performed well over the other mentioned higher-order approaches.

A Staggered Fourth-Order Accurate Explicit Finite Difference Scheme for the Time-Domain Maxwell's Equations

In this article, a family of high order accurate compact finite difference scheme for obtaining the approximate solution values of mildly nonlinear elliptic boundary value problems in three-space dimensions has been developed. The discretization formula is developed on a non-uniform meshes, which helps in resolving boundary and/or interior layers. The scheme involves 27 points single computational cell to achieve high order truncation errors. The proposed scheme has been applied to solve convection–diffusion equation, Helmholtz equation and nonlinear Poisson’s equation. A detailed convergence theory for the new compact scheme has been proposed using irreducible and monotone property of the iteration matrix. Numerical results show that the new compact scheme exhibit better performance in terms of \(l^\infty \)- and \(l^2\)-error of the exact and approximate solution values.

In this paper we design high-order accurate and stable finite difference schemes for the initial–boundary–value problem associated with the magnetic induction equation with resistivity. We use summation-by-parts finite difference operators to approximate spatial derivatives and a simultaneous approximation term technique for implementing boundary conditions. The resulting schemes are shown to be energy stable. Various numerical experiments demonstrating both the stability and the high order of accuracy of the schemes are presented.

High-order implicit finite difference schemes for the two-dimensional Poisson equation

An interpolation based finite difference method on non-uniform grid for solving Navier–Stokes equations

High-order compact finite difference schemes for solving the regularized long-wave equation

In this paper, a new conservative high-order compact finite difference scheme is studied for the initial-boundary value problem of the generalized Rosenau-regularized long wave equation. We design new conservative nonlinear fourth-order compact finite difference schemes. It is proved by the discrete energy method that the compact scheme is uniquely solvable; we have the energy conservation and the mass conservation for this approach in discrete Sobolev spaces. The convergence and stability of the difference schemes are obtained, and its numerical convergence order is $O(\tau^{2}+h^{4})$ in the $L^{\infty}$ -norm. Furthermore, numerical results are given to support the theoretical analysis. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

A shock-detecting sensor for filtering of high-order compact finite difference schemes

In this paper, first-order and second-order accurate structure-preserving finite difference schemes are proposed for solving the Schrödinger- Boussinesq equations. The conservation of the discrete energy and mass of the present schemes are analytically proved. Numerical experiments are given to support the theoretical results. Numerical examples show the efficiency of the proposed scheme and the correction of the theoretical proofs