Mathematical Sciences Department Numerical Methods Seminar - Bo Dong, Umass Dartmouth

Title: A new conservative discontinuous Galerkin method via implicit penalization for the generalized KdV equation

Abstract: Korteweg-de Vries (KdV) type equations are well known for having soliton solutions that maintain their shape, speed, and energy over a long period of time, even after interacting with other solitons. Research on the solutions of KdV equations has opened doors to many areas in nonlinear mathematics and theoretical physics, with applications in fields such as quantum mechanics, plasma physics, and fluid dynamics. In this talk, we study the design, analysis, and implementation of a new conservative Discontinuous Galerkin (DG) method for simulating solitary wave solutions to the generalized KdV equation. A key feature of our method is the conservation of mass, energy, and Hamiltonian, which are conserved by exact solutions of all KdV equations. To our knowledge, this is the first DG method that simultaneously conserves all three quantities, a property critical for the accurate long-time evolution of solitary waves. Our novel approach is to introduce two stabilization parameters in the numerical flux as new unknowns, which allows us to enforce energy and Hamiltonian conservation in the numerical scheme. We prove the conservation properties of the scheme, which are further validated by numerical experiments. Our approach of achieving conservation properties by implicitly defining stabilization parameters, that are traditionally specified a priori, provides a general framework for developing physics-preserving numerical methods for other types of problems.