An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions

Abstract

We are interested in numerically solving the Hamilton-Jacobi (HJ) equations, which arise in optimal control and many other applications. Oftentimes, such equations are posed in high dimensions, and this poses great numerical challenges. This work proposes a class of adaptive sparse grid (also called adaptive multiresolution) local discontinuous Galerkin (DG) methods for solving Hamilton-Jacobi equations in high dimensions. By using the sparse grid techniques, we can treat moderately high dimensional cases. Adaptivity is incorporated to capture kinks and other local structures of the solutions. Two classes of multiwavelets are used to achieve multiresolution, which are the orthonormal Alpert's multiwavelets and the interpolatory multiwavelets. Numerical tests in up to four dimensions are provided to validate the performance of the method.

An adaptive multiresolution ultra-weak discontinuous Galerkin method for nonlinear Schrodinger equations

“…Specifically, the second derivative term u xx is treated implicitly to avoid the severe CFL time constraint, while the nonlinear source f (|u| 2 )u is treated explicitly for efficiency. The adaptive procedure follows the technique developed in [20,21] to determine the space V that evolves dynamically over time.…”

“…It is also related to the adaptive mesh refinement (AMR) technique [2,4], which adjusts the computational grid adaptively to track small scale features of the underlying problems and improves computational efficiency. As a continuation of our previous research for adaptive multiresolution (also called adaptive sparse grid) DG methods [19,21,22,20], this paper develops an adaptive multiresolution ultra-weak DG solver for NLS equations (1.1) and the coupled NLS equations. First, the Alperts multiwavelets are employed as the DG bases in the weak formulation, and then the interpolatory multiwavelets are introduced for efficiently computing nonlinear source which has been successfully applied to nonlinear hyperbolic conservation laws [21] and Hamilton-Jacobi equations [20].…”

“…As a continuation of our previous research for adaptive multiresolution (also called adaptive sparse grid) DG methods [19,21,22,20], this paper develops an adaptive multiresolution ultra-weak DG solver for NLS equations (1.1) and the coupled NLS equations. First, the Alperts multiwavelets are employed as the DG bases in the weak formulation, and then the interpolatory multiwavelets are introduced for efficiently computing nonlinear source which has been successfully applied to nonlinear hyperbolic conservation laws [21] and Hamilton-Jacobi equations [20]. We refer the readers to [19,21] for more details on the background of adaptive multiresolution DG methods.…”