Seminar: Numerical Methods for Partial Differential Equations
Wednesday, October 05, 2022 at 4:30pm to 5:30pmVirtual Event
Speaker: David I. Ketcheson (King Abdullah University)
Title: Invariant-preserving methods for dispersive wave equations
Many dispersive wave equations have a Hamiltonian structure, with important conserved quantities of both linear and nonlinear type. The accuracy of numerical discretizations over long times depends critically on the preservation of these invariants. I will present a class of invariant-preserving discretizations based on summation-by-parts in space and relaxation in time; these fully-discrete schemes conserve both linear and nonlinear invariants and can be either implicit or explicit. These schemes have been developed for a wide range of such equations, including Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. I will show examples demonstrating that the error for such schemes grows only linearly in time, whereas for general schemes the error grows quadratically. I will also show examples of non-dispersive hyperbolic systems where such invariant-preserving schemes lead to a similar (drastic) improvement in long-time accuracy. Finally, I will present some preliminary results of conserving multiple nonlinear invariants in multi-soliton solutions.