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Multi-Moment ADER-Taylor Methods for Systems of Conservation Laws with Source Terms in One Dimension

Wednesday 26th September 2012 - 12:25 to 12:50
INI Seminar Room 1
Session Title: 
Time Session
A new integration method combining the ADER time discretization with a multi-moment finite-volume framework is introduced. ADER runtime is reduced by performing only one Cauchy-Kowalewski (C-K) procedure per cell per time step, and by using the Differential Transform Method for high-order derivatives. Three methods are implemented: (1) single-moment WENO [WENO], (2) two-moment Hermite WENO [HWENO], and (3) entirely local multi-moment [MM-Loc]. MM-Loc evolves all moments, sharing the locality of Galerkin methods yet with a constant time step during p -refinement.

Five experiments validate the methods: (1) linear advection, (2) Burger's equation shock, (3) transient shallow-water (SW) , (4) steady-state SW simulation, and (5) SW shock. WENO and HWENO methods showed expected polynomial h -refinement convergence and successfully limited oscillations for shock experiments. MM-Loc showed expected polynomial h -refinement and exponential p -refinement convergence for linear advection and showed sub-exponential (yet super-polynomial) convergence with p -refinement in the SW case.

HWENO accuracy was generally equal to or better than a five-moment MM-Loc scheme. MM-Loc was less accurate than RKDG at lower refinements, but with greater h - and p -convergence, RKDG accuracy is eventually surpassed. The ADER time integrator of MM-Loc also proved more accurate with p -refinement at a CFL of unity than a semi-discrete RK analog of MM-Loc. Being faster in serial and requiring less frequent inter-node communication than Galerkin methods, the ADER-based MM-Loc and HWENO schemes can be spatially refined and have the same runtime, making them a competitive option for further investigation.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons