Jennifer J. Zhao

HIGH-ORDER COMPACT SCHEME WITH MULTIGRID LOCAL MESH REFINEMENT PROCEDURE FOR CONVECTION-DIFFUSION PROBLEMS

We derive a new fourth order compact finite difference scheme which allows different meshsize in different coordinate directions for the two-dimensional convection diffusion equation. A multilevel local mesh refinement strategy is used to deal with the local singularity problem. A corresponding multilevel multigrid method is designed to solve the resulting sparse linear system. Numerical experiments are conducted to show that the local mesh refinement strategy works well with the high order compact discretization scheme to recover high order accuracy for the computed solution. Our solution method is also shown to be effective and robust with respect to the level of mesh refinement and the anisotropy of the problems.

Iterative solution and finite difference approximations to 3D microscale heat transport equation

Numerical techniques are proposed to solve a 3D time dependent microscale heat transport equation. A second-order finite difference scheme in both time and space is introduced and the unconditional stability of the finite difference scheme is proved. A computational procedure is designed to solve the resulting sparse linear system at each time step with a few iterative methods and their performances are compared experimentally. Numerical experiments are presented to demonstrate the accuracy of the finite difference scheme and the efficiency of the proposed computational procedure.

The Alternating Segment Explicit-Implicit scheme for the dispersive equation

In this paper, we present the Alternating Segment Explicit-Implicit scheme for the dispersive equation. The scheme is unconditionally stable and is capable of parallelism. The numerical simulations show that it has better accuracy than that of some existing schemes.

Truncation error and oscillation property of the combined compact difference scheme

In this work, we study a sixth order combined compact difference scheme for a one dimensional convection diffusion equation. We derive the truncation error representation and analyze its oscillation property. Numerical experiments are performed to illustrate and support our analysis.

Convergence and Error Bound Analysis for the Space-Time CESE Method

In this work, we study the convergence behavior of a recently developed space-time conservation element and solution element method for solving conservation laws. In particular, we apply the method to a one-dimensional time-dependent convection-diffusion equation possibly with high Peclet number. We prove that the scheme converges and we obtain an error bound. This method performs well even for strong convection dominance over diffusion with good long-time accuracy. Numerical simulations are performed to verify the results.

A high-order parallel finite difference algorithm

To solve the heat equation on parallel computers, a high-order parallel finite difference algorithm is presented. In this procedure, dividing the space domain into several sub-domains, we calculate the interface values between sub-domains by the classical explicit scheme, then solve the interior values of sub-domains by the forth-order compact scheme in parallel. The stability bound of the procedure is derived to be 1+63 times that of the classical explicit scheme. And the convergence rate is proved to be of order three. Numerical examples show that this method has much better accuracy than other known methods.

Error bound analysis of finite-difference approximations for a class of nonlinear parabolic system in two-space dimensions

In this paper, we establish the error bound for a finite-difference approximation to solutions of a class of Nonlinear Parabolic System in the form . We assume that the Cauchy Data is in and of class BV. We show that the sup norm of the error is bounded by for positive time.

Approximation to a parabolic system modeling the thermoelastic contacts of two rods

In this article, we study a sequence of finite difference approximate solutions to a parabolic system, which models two dissimilar rods that may come into contact as a result of thermoelastic expansion. We construct theapproximatesolutionsbasedonasetoffinitedifferenceschemestothesystem, andwewillprovethatthe approximate solutions converge strongly to the exact solutions. Moreover, we obtain and prove rigorously the error bound, which measures the difference between the exact solutions and approximate solutions in a reasonable norm. c 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 1{25, 1998

Space—time CESE method for the convection equation

In the paper, we study the analytic convergence of the space-time conservation element and solution element method for smooth solutions of the convection equation. Stability of the method is limited by the CFL condition and a parameter,e,which controls the numerical dissipation. We show that the method converges under appropriate conditions on the mesh and on the parameter,e, by considering a one-dimensional convection equation. The main advantage of this second order scheme is that it is an explicit marching scheme that allows one to solve for both the function and its derivative at the same time with comparable accuracy.Numerical simulations are presented to verify the convergence results for both the linear convection equation and a non-linear conservation law.

Convergence of numerical solutions

It is known that the zero level set of the solution of the Allen-Cahn equation ut - △u =e-2(u - u3) approaches, as e→ 0, to the solution of the motion by mean curvature flow. In this paper, we consider the convergence of numerical solutions of a discretized Allen-Cahn equation. In particular,we show that if for some p > 1, the space mesh size △x satisfies △z = O(ep), and the time mesh size satisfies the Courant-Friedrich.