6533b7d4fe1ef96bd126305f

RESEARCH PRODUCT

Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound

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We consider 3-point numerical schemes, that resolve scalar conservation laws, that are oscillatory either to their dispersive or anti-diffusive nature. The spatial discretization is performed over non-uniform adaptively redefined meshes. We provide a model for studying the evolution of the extremes of the oscillations. 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 behavior of the numerical schemes, decreases with time; hence proving that the overall scheme is TV Increase-Decreasing (TVI-D). We finally provide numerical evidence supporting the analytical results that exhibit the stabilization properties of the mesh adaptation technique. 1. Outline Mesh adaptation techniques have been employed by several authors in the past. It is worth noting the seminal works [DD87, For88, HH83, Luc85, Luc86, TT03], where several properties of mesh adaptation were studied. It was noticed in [AKM01, AMT04, AMS10, Sfa09] that proper use of non-uniform adaptively redefined meshes is capable of taming oscillations; hence improving the stability properties of the numerical schemes. To study the stabilization properties of mesh adaptation techniques we analyze the effect they have, on the oscillations that oscillatory/unstable numerical schemes produce. The setting is the one-dimensional scalar Riemann problem: ut+f(u)x = 0, x ∈ [a, b], with the flux function f being smooth and convex. For initial conditions we consider the single jump u0(x) = X[0,x0](x) with x, x0 ∈ (0, 1). We discretize spatially over a non-uniform adaptively redefined mesh. The mesh adaptation and the time evolution of the numerical solution are combined into the Main Adaptive Scheme: Definition 1.1 (Main Adaptive Scheme (MAS)). Given, at time step n, the mesh M x = {a = x1 < · · · < xN = b} and the approximations U = {u1 , . . . , uN}, the steps of the (MAS) are: 1. (Mesh Reconstruction). Construct new mesh: M x = {a = x 1 < · · · < x N = b}. 2. (Solution Update). Use the old mesh M x the approximations U n and the new mesh M x : Received by the editor September 19, 2009 and in revised form, September 2, 2011. 2010 Mathematics Subject Classification. Primary 65–XX. c ©2012 American Mathematical Society Reverts to public domain 28 years from publication