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Implicit for local effects and explicit for nonlocal effects is unconditionally stable. (English) Zbl 1085.76046

Summary: A combination of implicit and explicit timestepping is analyzed for a system of ODEs motivated by ones arising from spatial discretizations of evolutionary partial differential equations. Loosely speaking, the method we consider is implicit in local and stabilizing terms in the underlying PDE and explicit in nonlocal and unstabilizing terms. Unconditional stability and convergence of the numerical scheme are proved by the energy method and by algebraic techniques. This stability result is surprising because usually, when different methods are combined, the stability properties of the least stable method plays a determining role in the combination.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76R99 Diffusion and convection
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65L07 Numerical investigation of stability of solutions to ordinary differential equations
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