Difference Method For Parabolic Equations Research Articles

Stability analysis of inverse Lax–Wendroff boundary treatment of high order compact difference schemes for parabolic equations

In this paper, we study the stability of a numerical boundary treatment of high order compact finite difference methods for parabolic equations. The compact finite difference schemes could achieve very high order accuracy with relatively small stencils. To match the convergence order of the compact schemes in the interior domain, we take the simplified inverse Lax–Wendroff (SILW) procedure (Tan et al., 2012; Li et al., 2017) as our numerical boundary treatment. The third order total variation diminishing (TVD) Runge–Kutta method (Shu and Osher, 1988) is taken as our time-stepping method in the fully-discrete case. Two analysis techniques are adopted to check the algorithm’s stability, one is based on the Godunov–Ryabenkii theory, and the other is the eigenvalue spectrum visualization method (Vilar and Shu, 2015). Both the semi-discrete and fully-discrete cases are investigated, and these two different analysis techniques yield consistent results. Several numerical experimental results are shown to validate the theoretical results.

Difference methods for parabolic equations with Robin condition

Classical solutions of nonlinear second-order partial differential functional equations of parabolic type with the Robin condition are approximated in the paper by solutions of associated boundedness-preserving implicit difference functional equations. It is proved that the discrete solutions uniquely exist, they are uniformly bounded with respect to meshes and the numerical method is convergent and stable. We also find the error estimate and its asymptotic behavior. The properties of some auxiliary nonlinear discrete recurrent equations are showed. The proofs are based on the comparison technique and the Banach fixed-point theorem.

Research on the Forward Euler Difference Method for Parabolic Equations

This article is devoted to the forward Euler difference method for the parabolic equation. In this paper, a forward Euler difference scheme is derived. It is shown that the forward Euler difference scheme is convergence and stability. Moreover, a numerical experiment is conducted to illustrate the theoretical results of the presented method.

Cell-centered finite difference method for parabolic equation

We develop two cell-centered finite difference schemes for parabolic problems on quadrilateral grids. One scheme is backward Euler scheme with first order accuracy in time increment while the other is Crank–Nicolson scheme with second order accuracy in time increment. The method is based on the lowest order Brezzi–Douglas–Marini (BDM) mixed finite element method. A quadrature rule gives a block-diagonal mass matrix and allows for local velocity elimination, which leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical results indicate first-order convergence in spacial meshsize both for pressures and for subedge flues. Numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis.

Large time step maximum norm regularity of L-stable difference methods for parabolic equations

We establish almost sharp maximum norm regularity properties with large time steps for L-stable finite difference methods for linear second-order parabolic equations with spatially variable coefficients. The regularity properties for first and second spatial differences of the numerical solution mimic those of the continuous problem, with logarithmic factors in second differences. The regularity results for the inhomogeneous problem imply that the uniform rate of convergence of the numerical solution and its differences is controlled only by the maximum norm of the local truncation error.

ON NEW ADDITIVE DIFFERENCE METHOD FOR PARABOLIC EQUATIONS

A new approach is suggested to the construction of conservative difference schemes for multidimensional second-order parabolic equations containing divergent first-order differential forms. These schemes hold the property of having fixed sign for non-negative solutions. The convergence of schemes is proved in the grid norm L2 at the rate O(τ + h2).

Stability of High Order Accurate Difference Methods for Parabolic Equations with Boundary Conditions

We consider high order accurate difference schemes for second order parabolic equations in the quarter-plane $x \geqq 0,\, t \geqq 0$. We discuss the stability of these schemes for the mixed initial boundary value problem when particular high order accurate discrete boundary conditions are used.

On Difference Methods for Parabolic Equations and Alternating Direction Implicit Methods for Elliptic Equations

Some aspects are discussed of the development of the theory of difference approximations to parabolic equations in its relation to the basic development of difference methods for hyperbolic and elliptic equations by Courant, Friedrichs and Lewy, Math. Ann. 100, 32 (1928). The present paper also deals with the related problem of establishing the convergence of alternating direction implicit methods for elliptic problems.