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Parallel finite difference schemes with high-order accuracy and unconditional stability for solving parabolic equations are presented. The schemes are based on domain decomposition method, i.e., interface values between subdomains are computed by the explicit scheme; interior values are computed by the implicit scheme. The numerical stability and error are derived in the H^1 norm in one dimensional case. Numerical results of both one and two dimensions examining the stability, accuracy, and parallelism of the procedure are also presented.

A new derivation of the conservative domain decomposition procedure for solving the parabolic equation is presented. In this procedure, fluxes at subdomain interfaces are calculated from the solution at the previous time level, then these fluxes serve as Neumann boundary data for implicit, block-centered discretization in the subdomain. The unconditional stability and the second-order accuracy of solution values as well as fluxes are proved. Numerical results examining the stability, accuracy, and parallelism of the procedure are also presented.

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.

In this paper, a higher-order alternating group explicit scheme for the diffusion equation is developed. The scheme has fourth-order truncation error approximately. The numerical simulations show that the new scheme can provide more accurate solutions. A discussion on the numerical stability of the scheme is also included.

A finite difference domain decomposition algorithm (DDA) for solving the heat equation in parallel is presented. In this procedure, interface values between subdomains are calculated by the group explicit formula, whereas interior values of subdomains are determined by the classical implicit scheme. The stability and convergence for this DDA are proved. The stability bound of the procedure is derived to be eight times that of the classical explicit scheme. Though the truncation error at the interface is O(τ + h), L 2-error is proved to be O(τ + h 2). Numerical examples confirm the second-order convergence and indicate that the stability condition is sharp. A comparison of the numerical errors of this procedure with other known methods is also included.

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.

A new difference scheme with a higher-order discrete invariant for the KdV equation is proposed. Comparisons between the scheme and several other utilized schemes show that the new scheme is more effective in restraining nonlinear numerical instability.

Here the strategy of the group explicit method is applied to the numerical solution of a linear dispersive equation. An alternating group explicit method (AGE) for the equation with the periodic boundary condition is derived. The stability of the method is discussed. A comparison of the accuracy of this method with standard known methods is also included.

In this paper the parallel difference schemes with variable time steplengths for heat equations are studied. The absolute stability of the constructed schemes are proved. The numerical results of the alternating segment explicit-implicit (ASE-I) scheme and the alternating group explicit (AGE) scheme are presented. They show that the variable time steplength scheme provides good approximate solution that is more accurate than the solution of the equal time steplength scheme, and the former can be obtained with less computational effort than the latter.

In this paper, a difference scheme for the periodic initial-boundary problem of the coupled KdV Equation is given. The scheme keeps the first two conserved quantities which the differential equation possesses. The catch-ran iterative method is used to solve the difference equations. The numerical simulation exhibits the existence of two-soliton solutions.

In this paper, the convergence and stability of the explicit scheme and the weakly implicit scheme for a class of generalized KdV equation are proved. The sharp stability condition is obtained.