Analysis of Fully Discrete Approximations for Dissipative Systems and Application to Time-Dependent Nonlocal Diffusion Problems
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In this paper we first present stability and error analysis of the fully discrete numerical schemes for general dissipative systems, in which the implicit Runge–Kutta (IRK) method is adopted for time integration. Under suitable conditions on the IRK time stepping method that we refer as the total stability, a priori error estimates can be simultaneously obtained. Then we apply such time-marching techniques and analysis framework to one-dimensional time-dependent nonlocal diffusion problems, together with the discontinuous Galerkin method being used for spatial discretization. Unconditional stability of approximations of both primal and auxiliary variables and the priori error estimates for the corresponding fully discrete systems are proved, and the results indicate the schemes are asymptotically compatible. In addition, long time asymptotic behavior of the approximate solutions is also investigated. Various numerical experiments are finally performed to verify the theoretical results.
KeywordsFully discrete Strong stability Implicit Runge–Kutta Nonlocal diffusion Asymptotic compatibility
Mathematics Subject Classification65M60 65R20 65L06 65L07 45A05
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