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PDE Seminar

Linearized and nonlinear numerical stability for nonlinear PDEs


The theoretical issue of numerical stability and convergence analysis for a wide class of nonlinear PDEs is discussed in this talk. For most standard numerical schemes to certain nonlinear PDEs, such as the semi-implicit schemes for the viscous Burgers’ equation, a direct maximum norm analysis for the numerical solution is not available. In turn, a linearized stability analysis, based on an a-priori assumption for the numerical solution, has to be performed to make the local in time stability and convergence analysis go through. The linearized stability analysis usually requires a mild constraint between the time step and spatial grid sizes, therefore such a numerical stability is conditional. Instead, if a nonlinear numerical analysis could be directly derived, such as the convex splitting schemes for a class of gradient flows, a bound for the numerical solution becomes available, as a result of the energy stability. Therefore, the stability and convergence for these numerical schemes turn out to be unconditional..