Adaptive Mesh Reconstruction Total Variation Bound

We consider 3-point numerical schemes for scalar Conservation Laws, that are oscillatory either to their dispersive or anti-diffusive nature. Oscillations are responsible for the increase of the Total Variation (TV); a bound on which is crucial for the stability of the numerical scheme. It has been noticed ([AKM01], [AMT04], [AMS08]) that the use of non-uniform adaptively redefined meshes, that take into account the geometry of the numerical solution itself, is capable of taming oscillations; hence improving the stability properties of the numerical schemes. In this work we provide a model for studying the evolution of the extremes over nonuniform adaptively redefined meshes. Based on this model we prove that proper mesh reconstruction is able to control the oscillations; we provide bounds for the Total Variation (TV) of the numerical solution. We moreover prove under more strict assumptions that the increase of the TV -due to the oscillatory behaviour of the numerical schemesdecreases with time; hence proving that the overall scheme is TV Increase-Decreasing (TVI-D). 1 Outline We study the scalar Conservation Law in one space variable ut + f(u)x = 0, x ∈ [0, 1], t ∈ [0, T ] where the flux function f is considered to be smooth and convex. The initial condition throughout this work is: u0(x) = { 1, x ≤ x0 0, x > x0 , for x0 ∈ (0, 1) We discretise spatially over a non-uniform, adaptively redefined mesh, M = {0 = x0 < x n 1 < · · · < x n N = 1} The manipulation of the non-uniform mesh and the time evolution are combined into the Main Adaptive Scheme (MAS): Definition (Main Adaptive Scheme (MAS)). Given, at time step n, the mesh M x = {a = x1 < · · · < x n N = b} and the approximations U n = {u1 , . . . , u n N}, the steps of the (MAS) are as follows: University of Vienna, 1090 Vienna, Austria