Stability of operator splitting methods for systems with indefinite operators: reaction-diffusion systems

摘要: In this paper numerical results are reviewed [D.L. Ropp, J.N. Shadid, C.C. Ober, Studies of the accuracy time integration methods for reaction-diffusion equations, J. Comput. Phys. 194 (2) (2004) 544-574] that demonstrate common second-order operator-splitting can exhibit instabilities when integrating Brusselator equations out to moderate times about seven periods oscillation. These manifested as high wave number spatial errors. paper, we further analyze problem, and present a theorem stability applied linear with indefinite reaction terms which controls both low instabilities. A corollary shows if L-stable used diffusion term instability will be controlled more easily. absence L-stability, an additional step condition suppresses modes appears guarantee convergence at asymptotic order method. Numerical model problem confirm theory, agree well.