A characteristic mapping method for dissipative evolution equations

Abstract

The Characteristic Mapping (CM) method is a transport scheme which discretizes the Lagrangian flow map as a composition of spatial interpolants. This adaptive approximation technique has demonstrated novel resolution properties and the ability to efficiently simulate the fine scales of inviscid turbulent flows. In this work, we construct an extension of this approach to the solution of dissipative problems, including the Navier-Stokes equations. The extension lifts a general scalar evolution equation to a path in a group of diffeomorphisms, where the semigroup approach of the CM method can be exploited. We give an interpretation of this lift through the geometric framework of optimal transport. We then introduce a semidirect product evolution technique to perform the time stepping of the map components. The map is evolved based on the semi-Lagrangian CM approach, and its log-Jacobian is evolved explicitly by an exponential integrator. The numerical scheme is demonstrated to be high-order accurate and is tested on a range of canonical nonlinear dissipative equations. By retaining the resolution properties of the CM method, the approach shows a promising ability to simulate turbulent fluid flows across all Reynolds numbers.