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An Adaptive Multiresolution Interior Penalty Discontinuous Galerkin Method for Wave Equations in Second Order Form

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Abstract

In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) discontinuous Galerkin (DG) methods for simulating scalar wave equations in second order form in space. The two key ingredients of the schemes include an interior penalty DG formulation in the adaptive function space and two classes of multiwavelets for achieving multiresolution. In particular, the orthonormal Alpert’s multiwavelets are used to express the DG solution in terms of a hierarchical structure, and the interpolatory multiwavelets are further introduced to enhance computational efficiency in the presence of variable wave speed or nonlinear source. Some theoretical results on stability and accuracy of the proposed method are presented. Benchmark numerical tests in 2D and 3D are provided to validate the performance of the method.

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Acknowledgements

We would like to thank Daniel Appelö for discussions on numerical examples of wave propagation, Qi Tang and Kai Huang for the assistance and discussion in code implementation.

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Juntao Huang

  2. Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, KS, 67260, USA

    Yuan Liu

  3. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 70409, USA

    Wei Guo

  4. School of Mathematics, Jilin University, Changchun, 130012, Jilin, China

    Zhanjing Tao

  5. Department of Mathematics, Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Yingda Cheng

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  1. Juntao Huang
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  2. Yuan Liu
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  3. Wei Guo
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Correspondence to Zhanjing Tao.

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Research supported in part by a grant from the Simons Foundation (426993, Yuan Liu). Research is supported by NSF Grant DMS-1830838, Research is supported by NSF Grants DMS-1453661 and DMS-1720023. Research is supported by NSFC Grant 12001231 (Zhanjing Tao).

Interpolation Basis Functions

Interpolation Basis Functions

For completeness of the paper, we present details of the multiresolution interpolation basis functions, which are first introduced in [37]. We will first focus on the case in which the interpolation points are imposed in the inner domain, as implemented in Table 5. Then we discuss the case in which the points includes the cell interface points. Here, we only discuss the case when \(M=4\) and \(M=5\). For \(M=1,2,3\), we refer readers to the appendix in [27].

The basis functions in \(\tilde{W}_1\) are piecewise polynomials on \(I_l:=(0,\frac{1}{2})\) and \(I_r:=(\frac{1}{2},1)\). Note that the functions may be discontinuous at the interface \(x=1/2\), thus \(I_l\) and \(I_r\) are both defined to be open intervals. The basis functions in \(\tilde{W}_1\) in this paper are all supported on one half interval \(I_l\) or \(I_r\) and vanish on the other half. For simplicity, we will only declare the function on its support. For example, \(\psi _0(x)|_{I_r}\) gives the definition of \(\psi _0\) on \(I_r\) and indicates that \(\psi _0\) vanishes on \(I_l\).

1.1 Interpolation Points in the Inner Domain

1.1.1 \(M=4\)

The interpolation points are

$$\begin{aligned} \tilde{X}_0 = \left\{ \frac{1}{6},\frac{7}{24},\frac{1}{3},\frac{7}{12},\frac{2}{3} \right\} , \quad \tilde{X}_1 = \left\{ \frac{1}{12},\frac{7}{48},\frac{31}{48},\frac{19}{24},\frac{5}{6}\right\} . \end{aligned}$$

The basis functions in \(\tilde{W}_0^4\) and \(\tilde{W}_1^4\) are

$$\begin{aligned} \begin{array}{ll} \phi _0(x) = \frac{4}{45} (3 x-2) (3 x-1) (12 x-7) (24 x-7), \\ \phi _1(x) = -\frac{512}{189}(3 x-2) (3 x-1) (6 x-1) (12 x-7), \\ \phi _2(x) = \frac{1}{3} (3 x-2) (6 x-1) (12 x-7) (24 x-7), \\ \phi _3(x) = -\frac{32}{105}(3 x-2) (3 x-1) (6 x-1) (24 x-7), \\ \phi _4(x) = \frac{1}{27} (3 x-1) (6 x-1) (12 x-7) (24 x-7). \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l} \psi _{0}(x)|_{I_l}= \frac{8}{45} (3 x-1) (6 x-1) (24 x-7) (48 x-7), \\ \psi _{1}(x)|_{I_l}= -\frac{1024}{189} (3 x-1) (6 x-1) (12 x-1) (24 x-7), \\ \psi _{2}(x)|_{I_r}= -\frac{1024}{189} (3 x-2) (6 x-5) (12 x-7) (24 x-19), \\ \psi _{3}(x)|_{I_r}= -\frac{64}{105} (3 x-2) (6 x-5) (12 x-7) (48 x-31), \\ \psi _{4}(x)|_{I_r}= \frac{2}{27} (3 x-2) (12 x-7) (24 x-19) (48 x-31) \end{array} \end{aligned}$$

1.1.2 \(M=5\)

The interpolation points are

$$\begin{aligned} \tilde{X}_0 = \left\{ \frac{1}{12},\frac{1}{6},\frac{7}{24},\frac{1}{3},\frac{7}{12},\frac{2}{3} \right\} , \quad \tilde{X}_1 = \left\{ \frac{7}{48},\frac{1}{24},\frac{31}{48},\frac{19}{24},\frac{5}{6},\frac{13}{24} \right\} . \end{aligned}$$

The basis functions in \(\tilde{W}_0^5\) and \(\tilde{W}_1^5\) are

$$\begin{aligned} \begin{array}{ll} \phi _0(x) = \frac{1}{315} (-16) (3 x-2) (3 x-1) (6 x-1) (12 x-7) (24 x-7), \\ \phi _1(x) = \frac{4}{45} (3 x-2) (3 x-1) (12 x-7) (12 x-1) (24 x-7), \\ \phi _2(x) = -\frac{1024}{945}(3 x-2) (3 x-1) (6 x-1) (12 x-7) (12 x-1), \\ \phi _3(x) = \frac{1}{9} (3 x-2) (6 x-1) (12 x-7) (12 x-1) (24 x-7), \\ \phi _4(x) = -\frac{16}{315} (3 x-2) (3 x-1) (6 x-1) (12 x-1) (24 x-7), \\ \phi _5(x) = \frac{1}{189} (3 x-1) (6 x-1) (12 x-7) (12 x-1) (24 x-7), \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l} \psi _{0}(x)|_{I_l}= -\frac{2048}{945} (3 x-1) (6 x-1) (12 x-1) (24 x-7) (24 x-1), \\ \psi _{1}(x)|_{I_l}= -\frac{32}{315} (3 x-1) (6 x-1) (12 x-1) (24 x-7) (48 x-7), \\ \psi _{2}(x)|_{I_r}= -\frac{2048}{945} (3 x-2) (6 x-5) (12 x-7) (24 x-19) (24 x-13), \\ \psi _{3}(x)|_{I_r}= -\frac{32}{315} (3 x-2) (6 x-5) (12 x-7) (24 x-13) (48 x-31), \\ \psi _{4}(x)|_{I_r}= \frac{2}{189} (3 x-2) (12 x-7) (24 x-19) (24 x-13) (48 x-31), \\ \psi _{5}(x)|_{I_r}= -\frac{32}{315} (3 x-2) (6 x-5) (12 x-7) (24 x-19) (48 x-31) \end{array} \end{aligned}$$

1.2 Interpolation Points with the Interface Points

1.2.1 \(M=4\)

The interpolation points are

$$\begin{aligned} \tilde{X}_0 = \left\{ 0^+,\left( \frac{1}{4}\right) ^-, \left( \frac{1}{2}\right) ^-, \left( \frac{3}{4}\right) ^-, 1^- \right\} , \quad \tilde{X}_1 = \left\{ \left( \frac{1}{8}\right) ^-, \left( \frac{3}{8}\right) ^-, \left( \frac{1}{2}\right) ^+, \left( \frac{5}{8}\right) ^-, \left( \frac{7}{8}\right) ^- \right\} . \end{aligned}$$

The basis functions in \(\tilde{W}_0^4\) and \(\tilde{W}_1^4\) are

$$\begin{aligned} \begin{array}{ll} \phi _0(x) = \frac{1}{3} (x-1) (2 x-1) (4 x-3) (4 x-1), \\ \phi _1(x) = -\frac{16}{3} (x-1) x (2 x-1) (4 x-3), \\ \phi _2(x) = 4 (x-1) x (4 x-3) (4 x-1), \\ \phi _3(x) = -\frac{16}{3} (x-1) x (2 x-1) (4 x-1), \\ \phi _4(x) = \frac{1}{3} x (2 x-1) (4 x-3) (4 x-1). \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l} \psi _{0}(x)|_{I_l}= -\frac{32}{3} x (2 x-1) (4 x-1) (8 x-3), \\ \psi _{1}(x)|_{I_l}= -\frac{32}{3} x (2 x-1) (4 x-1) (8 x-1), \\ \psi _{2}(x)|_{I_r}= \frac{2}{3} (x-1) (4 x-3) (8 x-7) (8 x-5), \\ \psi _{3}(x)|_{I_r}= -\frac{32}{3} (x-1) (2 x-1) (4 x-3) (8 x-7), \\ \psi _{4}(x)|_{I_r}= -\frac{32}{3} (-32) (x-1) (2 x-1) (4 x-3) (8 x-5) \end{array} \end{aligned}$$

1.2.2 \(M=5\)

The interpolation points are

$$\begin{aligned} \tilde{X}_0 = \left\{ 0^+,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}, 1^- \right\} , \quad \tilde{X}_1 = \left\{ \frac{1}{10},\frac{3}{10},\left( \frac{1}{2}\right) ^-,\left( \frac{1}{2}\right) ^+,\frac{7}{10},\frac{9}{10} \right\} . \end{aligned}$$

The basis functions in \(\tilde{W}_0^5\) and \(\tilde{W}_1^5\) are

$$\begin{aligned} \begin{array}{ll} \phi _0(x) = -\frac{1}{24} (x-1) (5 x-4) (5 x-3) (5 x-2) (5 x-1), \\ \phi _1(x) = \frac{25}{24} (x-1) x (5 x-4) (5 x-3) (5 x-2), \\ \phi _2(x) = -\frac{25}{12} (x-1) x (5 x-4) (5 x-3) (5 x-1), \\ \phi _3(x) = \frac{25}{12} (x-1) x (5 x-4) (5 x-2) (5 x-1), \\ \phi _4(x) = -\frac{25}{24} (x-1) x (5 x-3) (5 x-2) (5 x-1), \\ \phi _5(x) = \frac{1}{24} x (5 x-4) (5 x-3) (5 x-2) (5 x-1), \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l} \psi _{0}(x)|_{I_l}= \frac{25}{3} x (2 x-1) (5 x-2) (5 x-1) (10 x-3), \\ \psi _{1}(x)|_{I_l}= \frac{50}{3} x (2 x-1) (5 x-2) (5 x-1) (10 x-1), \\ \psi _{2}(x)|_{I_r}= \frac{1}{3} x (5 x-2) (5 x-1) (10 x-3) (10 x-1), \\ \psi _{3}(x)|_{I_r}= -\frac{1}{3} (x-1) (5 x-4) (5 x-3) (10 x-9) (10 x-7), \\ \psi _{4}(x)|_{I_r}= -\frac{50}{3} (x-1) (2 x-1) (5 x-4) (5 x-3) (10 x-9), \\ \psi _{5}(x)|_{I_r}= -\frac{25}{3} (x-1) (2 x-1) (5 x-4) (5 x-3) (10 x-7). \end{array} \end{aligned}$$

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Huang, J., Liu, Y., Guo, W. et al. An Adaptive Multiresolution Interior Penalty Discontinuous Galerkin Method for Wave Equations in Second Order Form. J Sci Comput 85, 13 (2020). https://doi.org/10.1007/s10915-020-01322-w

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