FoCM 2014 conference

Workshop C2 - Foundation of Numerical PDE's - Semi-plenary talk

December 19, 17:05 ~ 17:55 - Room A22

The f-wave propagation algorithm for hyperbolic PDEs

Randall J. LeVeque

Applied Mathematics Department University of Washington, USA   -

Finite volume methods for solving hyperbolic PDEs, including nonlinear conservation laws whose solutions contain shock waves, are often based on solving one-dimensional Riemann problems. The wave-propagation algorithms implemented in the Clawpack software package provide a very general and robust approach to defining high-resolution methods that exhibit second-order accuracy in smooth regions of the solution while avoiding nonphysical oscillations near discontinuities. This approach is easily applied also to linear hyperbolic problems that are not in conservation form, and can be extended to two or three space dimensions by the introduction of "transverse Riemann solvers". These algorithms have also been generalized to the so-called f-wave formulation, in which the flux difference between adjacent cells is decomposed as a linear combination of eigenvectors of suitable flux Jacobian matrices. This approach has advantages in many applications including nonlinear problems with spatially varying flux functions, which arise for example in nonlinear elasticity problems in heterogeneous media. The f-wave approach also allows incorporating source terms directly into the Riemann solver in a natural manner, which is essential for some balance laws where the solution sought is a small perturbation to a nontrivial steady state in which nonzero source terms balance the divergence of the flux. Using the shallow water equations to model the propagation of a tsunami across the ocean is an example where this approach has been critical.

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