The Splitting Crank–Nicolson Scheme with Intrinsic Parallelism for Solving Parabolic Equations

Abstract

In this paper, a splitting Crank–Nicolson (SC-N) scheme with intrinsic parallelism is proposed for parabolic equations. The new algorithm splits the Crank–Nicolson scheme into two domain decomposition methods, each one is applied to compute the values at (n + 1)th time level by use of known numerical solutions at n-th time level, respectively. Then, the average of the above two values is chosen to be the numerical solutions at (n + 1)th time level. The new algorithm obtains accuracy of the Crank–Nicolson scheme while maintaining parallelism and unconditional stability. This algorithm can be extended to solve two-dimensional parabolic equations by alternating direction implicit (ADI) technique. Numerical experiments illustrate the accuracy and efficiency of the new algorithm.

Numerical solution of the Burgers equation associated with the phenomena of longitudinal dispersion depending on time

“…Here are some examples. Xue et al (2020) relied on the new algorithm obtained by the precision of the Crank-Nicolson scheme to solve two-dimensional parabolic equations by alternating the implicit technique of direction. Kumar and Unny (1977) focussed his research on the application of Runge-Kutta methods to solve nonlinear partial differential equations using a specific fluid flow problem.…”