Hyperbolic conservation laws with source: solution by finite differences using log conformation

Abstract

We shall consider some important aspects from the numerical point of view, namely: I) treatment of convective terms; II) numerical treatment when we include source terms in analogy to what happens in the simulation of viscoelastic fluid flows. The demand for more robust techniques is still an important source of research, mainly because more complete equations motivate new developments, this is the case of an equation with source term. A naive look could induce that equations without source and with source are similar and therefore any numerical method designed to the first one could be easily applied to the second. Actually, this is not always valid. In a broader context, this situation is synthesized by the number of Weissemberg (We). Simply put, the larger We, more intense nature of the interactions of elastic and most decisive contribution to the flow term source. However, traditional methods commonly employed bump into "fuzzy" restrictions on the magnitude of We, on which much is discussed the cause: if restrictions are likely to be numerical or intrinsic of the viscoelastic model. This has annoyed and aroused the interest of several researchers in recent decades, constituting the so-called High Weissember Number Problem (HWNP).Recently, through log-conformation representation (CRL), better indication that the cause is numerical. Just as the numerical solution without source term need the correct balance for the convection, the source solution must take into consideration the balance between the convective terms of nonlinear nature (which generates the hyperbolic character) and elastic source term (which introduces a stiff character of the equation in part). Effectively, this is the summit we want to achieve in this work, analyzing and reproducing the mechanism associated with HWNP through the study of simplified equations. A simple problem which allows extensive study of these aspects is the Burgers equation with source. Thus, it is possible the treatment of several important concepts in the analysis of numerical algorithms, which, being studied for this simple case, can be applied to more complex equations. Thus, the study will cover the classical methods for finite difference essentially non-oscillatory TVD type and its properties and, following the analysis of properties that change when we go from no source for a source. This includes a detailed study of the log-conformation representation and its implementation. (AU)