Hermite methods for hyperbolic initial-boundary value problems

Goodrich, John; Hagstrom, Thomas; Lorenz, Jens

doi:10.1090/S0025-5718-05-01808-9

Hermite methods for hyperbolic initial-boundary value problems
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by John Goodrich, Thomas Hagstrom and Jens Lorenz PDF

Math. Comp. 75 (2006), 595-630 Request permission

Abstract:

We study arbitrary-order Hermite difference methods for the numerical solution of initial-boundary value problems for symmetric hyperbolic systems. These differ from standard difference methods in that derivative data (or equivalently local polynomial expansions) are carried at each grid point. Time-stepping is achieved using staggered grids and Taylor series. We prove that methods using derivatives of order $m$ in each coordinate direction are stable under $m$-independent CFL constraints and converge at order $2m+1$. The stability proof relies on the fact that the Hermite interpolation process generally decreases a seminorm of the solution. We present numerical experiments demonstrating the resolution of the methods for large $m$ as well as illustrating the basic theoretical results.

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