Sparse + low-energy decomposition for viscous conservation laws

For viscous conservation laws, solutions contain smooth but high-contrast features, which require the use of fine grids to properly resolve. On coarse grids, these high-contrast jumps resemble shocks rather than their true viscous profiles, which could lead to issues in the numerical approximation of their underlying dynamics. In many cases, the equations of motion emit traveling wave solutions which can be used to represent the viscous profiles analytically. The traveling wave solution can be thought of as a lower dimensional representation of the motion, since they contain information from the evolution equation, but are constant along certain time-space curves. Using a parameterized basis involving the traveling waves, along with the sparse + low-energy decompositions found in imaging sciences, we propose an approximation to viscous conservation laws which separates the coarse smooth component from the sharp fine one. Our method provides an appropriate approximation to the solution on a coarse grid, thereby accurately under-resolving the viscous profile. This is similar to the philosophy of shock capturing methods, in the sense that we want to capture the viscous front without needing to resolve the profile. Theoretical results on the consistency of our method are shown in general. We provide several computational examples for convex and non-convex fluxes.