A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations
Abstract
:1. Introduction
2. General Formulation of Path-Conservative ADER-DG Method
- For every and ,
- For every , , and ,
2.1. Applications to 1D SWEs
2.1.1. Notations and Solution Space
2.1.2. Construction of 1D Path-Conservative ADER-DG Method
- The Osher jump term:
- The Roe jump term:
- -
- For each , the matrix owns m different real eigenvalues
- -
- The compatibility property
- -
- For arbitrary , , the matrix satisfies the below requirement
2.1.3. The DT Procedure
2.1.4. The Slope Limiter
2.1.5. Implementation Details of 1D Path-Conservative ADER-DG Method
3. Extension to 2D System
4. Numerical Results
4.1. 1D System
4.1.1. WB Property Test
4.1.2. Accuracy Test
4.1.3. Perturbations of a Steady State Water Flow
4.1.4. The Dam Break Problem over a Rectangular Bottom
4.1.5. Steady Flow over a Hump
- Case A: the transcritical flow without a shockat the upstream boundary; on the downstream one.
- Case B: the transcritical flow with a shockon the upstream boundary; on the downstream one.
- Case C: the subcritical flowon the upstream boundary; on the downstream one.
4.1.6. The Dam Break Problem over a Step
- Case AOver time, this example produces a rarefaction moving to the left and a shock moving to the right.
- Case BThe initial conditions are
- Case C
- Case DThe initial data are
4.2. 2D System
4.2.1. WB Property Test
4.2.2. A Small Perturbation of a 2D Steady State Flow
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Algorithm A1: The algorithm of the DT procedure for 1D SWEs |
|
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Original Function | Transformed Function |
---|---|
Precision | Error | Error | ||||
---|---|---|---|---|---|---|
Single | ||||||
Double |
Precision | ||||||
---|---|---|---|---|---|---|
Single | ||||||
Double |
Cells | h | |||
---|---|---|---|---|
Order | Order | |||
25 | ||||
50 | 2.33 | 2.43 | ||
100 | 2.41 | 2.51 | ||
200 | 2.56 | 2.66 | ||
400 | 2.70 | 2.70 | ||
800 | 2.83 | 2.86 | ||
1600 | 2.93 | 2.90 | ||
3200 | 3.01 | 2.99 |
Precision | Error | ||
---|---|---|---|
Single | |||
Double |
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Zhao, X.; Wang, B.; Li, G.; Qian, S. A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations. Mathematics 2024, 12, 2601. https://doi.org/10.3390/math12162601
Zhao X, Wang B, Li G, Qian S. A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations. Mathematics. 2024; 12(16):2601. https://doi.org/10.3390/math12162601
Chicago/Turabian StyleZhao, Xiaoxu, Baining Wang, Gang Li, and Shouguo Qian. 2024. "A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations" Mathematics 12, no. 16: 2601. https://doi.org/10.3390/math12162601