A New Splitting Method for Time-dependent Convection-dominated Diffusion Problems

TL;DR

This paper introduces a novel splitting method for efficiently solving time-dependent convection-dominated diffusion problems by separating convection and diffusion steps, enhancing stability and computational efficiency without tuning parameters.

Contribution

A new splitting scheme that combines explicit convection solving with implicit diffusion, extended to Navier-Stokes equations, avoiding stabilization parameter tuning.

Findings

Scheme is unconditionally stable with multistep enhancement.

Efficiently solves diffusion using existing preconditioned iterative methods.

Numerical results confirm stability and convergence.

Abstract

We present a new splitting method for time-dependent convection-dominated diffusion problems. The original convection diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. The scheme has the following nice features: the convection subproblem is solved explicitly and a multistep technique is introduced to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; the diffusion subproblem is always self-adjoint and coercive so that it can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme is then extended for Navier-Stokes equations, where the nonlinear convection is resolved by a linear explicit multistep scheme at the convection step, and only a generalized Stokes problem is needed to solve at the diffusion step with the resulting stiffness matrix being invariant in the time marching process. The new schemes are all free from tuning some stabilization parameters for the convection-dominated diffusion problems. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.