An adaptive multiresolution discontinuous Galerkin method with artificial viscosity for scalar hyperbolic conservation laws in multidimensions

Abstract

In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for scalar hyperbolic conservation laws in multidimensions. Compared with previous work for linear hyperbolic equations [25,26], a class of interpolatory multiwavelets are applied to efficiently compute the nonlinear integrals over elements and edges in DG schemes. The resulting algorithm, therefore can achieve similar computational complexity as the sparse grid DG method for smooth solutions. Theoretical and numerical studies are performed taking into consideration of accuracy and stability with regard to the choice of the interpolatory multiwavelets. Artificial viscosity is added to capture the shock and only acts on the leaf elements taking advantages of the multiresolution representation. Adaptivity is realized by auto error thresholding based on hierarchical surplus. Accuracy and robustness are demonstrated by several numerical tests.

An adaptive high-order piecewise polynomial based sparse grid collocation method with applications

“…In this paper, we consider general functions that are supported on the grid Ω N and are allowed to be discontinuous at the interface of Ω N , where N is a prescribed integer. This is particularly needed for the implementation of multiresolution DG scheme [17]. In definition (2.1), if the point x j i,n lands on the interface of Ω N , it should be defined either as the left or right limit point.…”

“…Now we consider several examples in UQ. Note that another application area is in the design of adaptive multiresolution DG methods, which has been considered in [17].…”

“…For function interpolation, it was verified that higher order methods perform better for smooth functions, and the Hermite interpolation methods provide a solution representation more concentrated towards singularities. In a separate work [17], we apply the collocation scheme to facilitate the computation of adaptive multiresolution DG scheme for nonlinear hyperbolic equations. It was found in [17] that the Hermite interpolation provides more stable numerical solution than Lagrange interpolation.…”

“…In a separate work [17], we apply the collocation scheme to facilitate the computation of adaptive multiresolution DG scheme for nonlinear hyperbolic equations. It was found in [17] that the Hermite interpolation provides more stable numerical solution than Lagrange interpolation. Another possible application of this work is to construct adaptive semi-Lagrangian schemes, which will be explored in the future.…”

“…Those techniques are not explored in this work. Rather, our main motivation is the design of adaptive multiresolution DG methods, which is considered separately in another work [17].…”