Adaptive multiresolution discontinuous Galerkin schemes for conservation laws: multi-dimensional case

Abstract

The concept of multiresolution-based adaptive DG schemes for non-linear one-dimensional hyperbolic conservation laws has been developed and investigated analytically and numerically in (Math Comp, doi:10.1090/S0025-5718-2013-02732-9, 2013). The key idea is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. The focus of the present work lies on the extension of the originally one-dimensional concept to higher dimensions and the verification of the choice for the threshold value by means of parameter studies performed for linear and non-linear scalar conservation laws.

Appendix: multiwavelets on the reference element \([0,1]^2\)

In this section we give some examples of multiwavelets which are constructed by Algorithm 1 and used for the computations in Sect. 5.

For a detailed description of the construction we refer to Müller (2013). To simplify their representation we use the following initialization in step (i) of Algorithm 1:

$$\begin{aligned}&\psi _{\lambda , i,0}({\mathbf x}) := \left\{ \begin{array}{l@{\quad }l} \left( {\mathbf x}-{\mathbf x}_c\right) ^i &{} \text {if } {\mathbf x}\in (0.5,1.0)^2 \\ -\left( {\mathbf x}-{\mathbf x}_c\right) ^i &{} \text {if } {\mathbf x}\in (0, 0.5)^2 \\ 0&{}\text {else} \end{array}\right. , \quad \\&\psi _{\lambda , i,1}({\mathbf x}) := \left\{ \begin{array}{l@{\quad }l} \left( {\mathbf x}-{\mathbf x}_c\right) ^i &{} \text {if } {\mathbf x}\in (0.5,1) \times (0,0.5) \\ -\left( {\mathbf x}-{\mathbf x}_c\right) ^i &{} \text {if } {\mathbf x}\in (0,0.5) \times (0.5,1)\\ 0&{}\text {else} \end{array}\right. , \\&\psi _{\lambda , i,2}({\mathbf x}) := \left\{ \begin{array}{l@{\quad }l} \left( {\mathbf x}-{\mathbf x}_c\right) ^i &{} \text {if } {\mathbf x}\in (0,0.5)^2 \cup (0,5,1)^2 \\ -\left( {\mathbf x}-{\mathbf x}_c\right) ^i &{} \text {if } {\mathbf x}\in (0.5,1) \times (0, 0.5) \cup (0,0.5) \times (0.5,1)\\ 0&{}\text {else} \end{array}\right. , \end{aligned}$$

with ${\mathbf x}_c = \left( 0.5, 0.5 \right) ^t$.

Since the construction is done on a reference element, the uniform boundedness of the multiwavelets in (3.8) is satisfied for any scaling of the multiwavelets on the reference element. The multiwavelets we use in our computations in Section 5 are normalized with respect to $L_2((0,1)^2)$ on the reference element and then shifted to the local cells in the grid.

In Tables 3 and 4, we list all multiwavelets for $p=1,2,3$. The multiwavelets are shown here without normalization to keep them clearly arranged. For simplicity we name the multiwavelets on the reference element $V_{\lambda } = (0,1)^2$ by $\hat{\psi }_{i}$. The multiwavelets are enumerated from $1$ to $3 \vert {\mathcal P}\vert$. Then a multiwavelet is uniquely defined by four polynomials:

$$\begin{aligned} \hat{\psi }_{i}(x,y) := \left\{ {\begin{array}{l@{\quad }l} \hat{\psi }_{i, 1}(x,y) &{} \text { if } (x,y) \in [0.5,1] \times [0.5,1],\\ \hat{\psi }_{i, 2}(x,y) &{} \text { if } (x,y) \in (0.5,1) \times (0,0.5),\\ \hat{\psi }_{i, 3}(x,y) &{} \text { if } (x,y) \in (0,0.5) \times (0,0.5),\\ \hat{\psi }_{i, 4}(x,y) &{} \text { if } (x,y) \in (0,0.5) \times (0.5,1),\\ 0&{} \text { else. } \end{array}}\right. \end{aligned}$$

Table 3 Wavelets for $p=1,2,3$—part I

Table 4 Wavelets for $p=1,2,3$—part II