Abstract
This paper proposes a new class of difference methods with intrinsic parallelism for solving the Burgers–Fisher equation. A new class of parallel difference schemes of pure alternating segment explicit-implicit (PASE-I) and pure alternating segment implicit-explicit (PASI-E) are constructed by taking simple classical explicit and implicit schemes, combined with the alternating segment technique. The existence, uniqueness, linear absolute stability, and convergence for the solutions of PASE-I and PASI-E schemes are well illustrated. Both theoretical analysis and numerical experiments show that PASE-I and PASI-E schemes are linearly absolute stable, with 2-order time accuracy and 2-order spatial accuracy. Compared with the implicit scheme and the Crank–Nicolson (C-N) scheme, the computational efficiency of the PASE-I (PASI-E) scheme is greatly improved. The PASE-I and PASI-E schemes have obvious parallel computing properties, which show that the difference methods with intrinsic parallelism in this paper are feasible to solve the Burgers–Fisher equation.
Highlights
The Intrinsic Parallel Difference Schemes of Burgers–Fisher EquationE points calculated at the same time layer are divided into J segments, which are recorded in the order S1, S2,
Introduction e BurgersFisher equation is a typical model for describing diffusion propagation and convection conduction
To obtain a parallel difference scheme with higher precision and more relaxed stability conditions, we study the parallel algorithm of the Burgers–Fisher equation from the point of view of the parallelization of traditional difference schemes
Summary
E points calculated at the same time layer are divided into J segments, which are recorded in the order S1, S2, . Every segment of the odd time layer is arranged from left to right in the order of “classical explicit-classical implicit-classical explicit.”. On the even time layer, the order of arrangement becomes “classical implicitclassical explicit-classical implicit.” e solution of each implicit segment relies on the calculation of the first or last point of the adjacent explicit segment to give its internal boundary value. For realizing the parallel computing of the PASE-I scheme, for i0 ≥ 0, we consider the calculation of the explicit segment point (i0 + i, n + 1), i 1, 2, . Compared with the classical implicit scheme, there are fewer ((J + 1)/2) equation systems in the odd layer and fewer ((J − 1)/2). Equation systems in the even layer. e computation is simple, and the parallel characteristic is obvious
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