Explicit numerical integrators that conserve energy or dissipate entropy

January 14, 2022
Many mathematical models are equipped with an energy that is conserved or an entropy that is known to change monotonically in time. Integrators that preserve these properties discretely are usually expensive, with the best-known examples being fully-implicit Runge-Kutta methods. I will present a modification that can be applied to any integrator in order to preserve such a structural property. The resulting method can be fully explicit, or (depending on the functional) may require the solution of a scalar algebraic equation at each step. I will present examples to show the effectiveness of these “relaxation” methods, and their advantages over fully implicit methods or orthogonal projection. Examples will include applications to compressible fluid dynamics, dispersive nonlinear waves, and Hamiltonian systems.