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Self-adaptive time integration of flux-conservative equations with sources. (English) Zbl 1093.65083
Summary: We present a novel, flux-conserving, asynchronous method for the explicit time integration of multi-scale, flux-conservative partial differential equations with source terms. Unlike the conventional explicit and implicit integration schemes, it is based on a discrete-event simulation paradigm, which describes time advance in terms of increments to physical quantities and causality rules rather than time stepping. This method exerts self-adaptive control over local update rates of solution by predicting and correcting changes to simulation variables in accordance with local physical scales. The discrete-event simulation paradigm is independent of the underlying spatial mesh and thus can be incorporated into block-structured and unstructured mesh refinement techniques.
The effectiveness and robustness of the new method is demonstrated on a number of one-dimensional, uniform mesh models based on diffusion-convection-reaction equations. The event-driven integration reduces numerical approximation errors due to large local time derivatives, prevents explosive numerical instabilities in locally super-Courant calculations and automatically reduces the CPU overhead associated with stiff terms and inactive regions in computation space.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
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