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

TL;DR

This paper introduces a sparse grid time-discontinuous Galerkin method with streamline diffusion for efficiently solving high-dimensional transport equations, demonstrating stability, convergence, and applicability up to six dimensions.

Contribution

The paper develops a novel sparse grid Galerkin method with streamline diffusion for high-dimensional transport equations, combining geometric flexibility and GPU acceleration.

Findings

Method is stable and convergent for complex geometries.

Effective in solving up to six-dimensional problems.

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