Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations, Part III. The Effects of Nonlinear Source Terms in Reaction-Convection Equations

Abstract

Abstract The objective of this paper is to investigate how well numerical steady-state solutions can mimic true steady-states of a model nonlinear reaction-convection. In order to assess the possible'sourcesoferrors,slowconvergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach, six different numerical aspects are addressed. First, spurious stable and unstable steady-state numerical solutions (numerically irrelevant solutions) can be independently introduced by spatial and temporal discretizations satisfying the same boundary condition and initial data. Second, the various numerical treatments of the reaction term can affect the stability of the spurious as well as the exact steady-state solutions. Third, the time discretization can destabilize the spurious stable steady-state numerical solutions that are introduced by the spatial discretizations or vice versa. Fourth, for a given finite difference method, the strength of the coefficient of the convection term and stiffness of the nonlinear source term can strongly affect the permissibility of spurious steady-state numerical solutions. Fifth, the possiblecause of convergence to a spurious steady-state and a suggestion to avoid spurious steady-states are discussed. Finally, the numerical phenomenon of incorrect propagation speeds of discontinuities may be linked to the existence of some spurious stable steady-state numerical solutions.