AMCS 348 Summation by Parts Operators for PDEs

The past decade has seen an explosion in popularity of developing methods of any order with the summation- by-parts (SBP) property. This is because the SBP framework offers a simple, yet powerful methodology for the design and analysis of modern algorithms for the solution of partial differential equations (PDEs). The summation-by-parts (SBP) concept was originally developed in the finite difference community with the goal of mimicking finite element energy analysis techniques. In recent years, this simple idea has been exponentially generalized enabling a unifying framework for the stability analysis of many spatial discretizations including finite difference, finite volume, flux reconstruction, and continuous/discontinuous Galerkin (FEM) methods on structured and unstructured polytope meshes for linear and nonlinear conservation laws on conforming and non-conforming grids. The most important consequence of SBP is that it naturally guides the path to stability and robustness as it mimics continuous stability analysis. The SBP concept provides a strong theoretical framework, that is discretization agnostic, for the analysis of existing schemes and the design of flexible high-order numerical approximations that are robust for complex multi-scale applications. The main topics of the course will be: introduction of SBP operators through the stability analysis of model problems equations (advection and advection-diffusion equations); construction of collocated Legendre-Gauss-Lobatto SBP operators; Hadamard formalism and extension of SBP operators to nonlinear PDEs; one element discretizations analysis and weak imposition of boundary conditions through the simultaneous-approximation terms technique; multi-element discretizations and analysis and imposition of weak interface coupling; discontinuous collocated Galerkin method and entropy stability for the Burgers’, compressible Euler and Navier-Stokes equations.




AMCS 252