Some Researches On Parallel Algorithm For Convection-Diffusion Equation

The convection-diffusion equation is usually used to describe situation, such as fluid motion, heat conduction, particle diffusion, etc. This equation therefore plays an important role in scientific and engineering computation. Among the finite difference methods for the numerical solutions of convection-diffusion equation , the classical explicit scheme is limited to the stability condition;the classical implicit scheme and the well-known Crank-Nicolson method is unconditionally stable ,but it requires to solve a large scale linear system which is no longer tridiagonal and difficult to directly solve on parallel computers .In recent years , Professor Evans and Zhang Baolin have brought forward a series of parallel algorithm for diffusion equations (two dimension) and convection-diffusion equation(one dimension or two dimension) ,such as the Alternating Group Explicit method (AGE),the Alternating Group Explicit-Implicit method (AGE-I), the Alternating Block Crank-Nicolson method (ABC-N). the Alternating Band Crank-Nicolson method (ABdC-N).Further more, Professor Wang Wenqia in Shan Dong University has advanced the Alternating Segment Crank-Nicolson method (ASC-N) for one-dimensional convection-diffusion equation .In this paper ,a series of parallel algorithm have been brought forward for one-dimensional Burgers equation and two-dimensional convection-diffusion equation . The content from chapter two to chapter five is the soul of this paper.For one-dimensional convection-diffusion equation, I put forward a new parallel algorithm of ASC-N method which is unconditionally stable and based on the saul'yev asymmetric schemes and Crank-Nicolson scheme in chapter two. The numerical experiments show that the parallel algorithm is both feasible and effective .For two-dimensional convection-diffusion equation, I put forward a new parallel algorithm of ABC-N method which is based on the saul'yev asymmetric schemes and Crank-Nicolson scheme in chapter three. The method also has the advantage of unconditional stability ,suitable for high performance parallel computer .The numerical solutions show that the method has quite better accuracy.The ABdC-N method which has the advantage of parallel computing , unconditional stability is based on the saul'yev asymmetric schemes and Crank-Nicolson scheme is constructed in chapter four .The method also can extend to the non-linear situation ,the numerical solutions make clear that the method has quite better accuracy.The ABE-I method for two-dimensional convection-diffusion equation is brought forward in chapter five .The method also has the advantage of unconditional stability , parallel computing yet and also can extend to the non-linear situation ,the numerical solutions indicate that the method has quite better accuracy.