The multistep finite difference fractional steps method for a class of viscous wave equations

Abstract

In this paper, a new multistep finite difference fractional method applicable to parallel arithmetic for viscous wave equation is proposed. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high-order difference operators and priori estimates are adopted. It is shown that the scheme is second-order in temporal and spacial direction in the l 2 norm. The new scheme is unconditionally stable for initial value by using the Von Neumann linear stability analysis. Experiments show that the new method is very efficient for solving viscous wave equation, which is of vital importance in life sciences.

A Convergence Analysis of a Multistep Method Applied to an Advection-Diffusion Equation in 1-D

“…Similar to the consistency analysis presented by Oliveira et al [ 1 ], we used the parametrization of the error as a function of time and space discretization. Additionally, the stability analysis follows a technique employed by Zhang [ 7 ]. In particular, the use of backward differentiation formulas did not improve the accuracy of the standard Hopmoc method, i.e., the BDFHM method is first-order accurate in space and time.…”

“…For example, Oliveira et al [ 1 ] demonstrated the stability of the Hopmoc method utilizing a spectral condition involving the values of and . Another example is a technique proposed by Zhang [ 7 ]. This technique is based on the decomposition of operators to prove the stability of a second order time multistep method, which was applied to the parallel solution of the wave equation.…”

“…In a finite-difference approximation, a solution of the finite-difference equation is expanded in a Fourier series. Thereby, this present stability analysis is based on the techniques presented by Zhang [ 7 ]. For clarity, we show Eqs.…”

“…( 4 ) and ( 5 ) again below. Thus, taking into account the strategy proposed by Zhang [ 7 ], consider two time semi-steps of the BDFHM method, and the difference operator , recalling that . Both time semi-steps ( 18 ) and ( 19 ) can be rewritten in function of as and Let’s be a auxiliary component, such that and .…”

“…This paper presents a convergence analysis of the BDFHM method. The consistency and stability analyses rely on the techniques employed by Oliveira et al [ 1 ] and Zhang [ 7 ], respectively.…”