Sparse grid time-discontinuous Galerkin method with streamline diffusion for transport equations

Abstract High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm efficiently solve the equation moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are tensor product of sparse grid space discontinuous piecewise polynomials time. Here, constructed upon nested multilevel finite element provide geometric flexibility. This results implicit time-stepping scheme which prove be stable convergent. If solution has additional mixed regularity, convergence 2 d -dimensional problem equals that one up logarithmic factors. For implementation, rely on representation grids as sum anisotropic full spaces. enables us store functions carry out computations sequence regular exploiting structure way existing libraries GPU acceleration can used. combination technique used preconditioner for iterative time strips. Numerical tests show works well problems six dimensions. Finally, also building block nonlinear Vlasov-Poisson equations.