AFRL-AFOSR-VA-TR-2015-0337 Entropy Viscosity and L1-based Approximations of PDEs: Exploiting Sparsity

Our goal is to develop robust numerical methods for solving mathematical models of nonlinear phenomena such as nonlinearconservation laws, advection-dominated multi-phase flows, and free-boundary problems, where shocks, fronts, and contactdiscontinuities are driving features and pose significant difficulties for traditional numerical methods. We have discovered thattime-dependent nonlinear conservation equations can be stabilized by using the so-called entropy viscosity method and weproposed to to investigate this new technique. We explored in detail the approximation properties of the entropy viscositymethod along the following directions. (i) New discretization methods including Discontinuous Galerkin and Lagrangianhydrodynamics; (ii) New fields of applications of the entropy viscosity concept, such as multiphase flows, using phase fieldtechniques. This novel robust approximation technique for solving nonlinear problems developing shock or sharp interfaceswill benefit every areas of science and engineering where controlling or dealing with this type of phenomenon is still anDISTRIBUTION A: Distribution approved for public release.