Entropy Viscosity and L1-based Approximations of PDEs: Exploiting Sparsity.

摘要

Our goal is to develop robust numerical methods for solving mathematical models of nonlinear phenomena such as nonlinear conservation laws, advection-dominated multi-phase flows, and free-boundary problems, where shocks, fronts, and contact discontinuities are driving features and pose significant difficulties for traditional numerical methods. We have discovered that time-dependent nonlinear conservation equations can be stabilized by using the so-called entropy viscosity method and we proposed to investigate this new technique. We explored in detail the approximation properties of the entropy viscosity method along the following directions. (i) New discretization methods including Discontinuous Galerkin and Lagrangian hydrodynamics; (ii) New fields of applications of the entropy viscosity concept, such as multiphase flows, using phase field techniques. This novel robust approximation technique for solving nonlinear problems developing shock or sharp interfaces will benefit every areas of science and engineering where controlling or dealing with this type of phenomenon is still an enormous challenge.