A well-balanced stable generalized Riemann problem scheme for shallow water equations using adaptive moving unstructured triangular meshes

• Published in: International journal for numerical methods in fluids Vol. 73; no. 3; pp. 266 - 283
• Main Authors: Zhou, Feng, Chen, Guoxian, Noelle, Sebastian, Guo, Huaicheng
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 30.09.2013; Wiley Subscription Services, Inc
• Subjects: Accuracy; adaptive unstructured meshes; generalized Riemann problem; Glass fiber reinforced plastics; hydrodynamic process; Lakes; Mathematical analysis; Mathematical models; Shallow water equations; Singularities; Topography; well-balanced scheme
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.3800

SUMMARYWe propose a well‐balanced stable generalized Riemann problem (GRP) scheme for the shallow water equations with irregular bottom topography based on moving, adaptive, unstructured, triangular meshes. In order to stabilize the computations near equilibria, we use the Rankine–Hugoniot condition to remove a singularity from the GRP solver. Moreover, we develop a remapping onto the new mesh (after grid movement) based on equilibrium variables. This, together with the already established techniques, guarantees the well‐balancing. Numerical tests show the accuracy, efficiency, and robustness of the GRP moving mesh method: lake at rest solutions are preserved even when the underlying mesh is moving (e.g., mesh points are moved to regions of steep gradients), and various comparisons with fixed coarse and fine meshes demonstrate high resolution at relatively low cost. Copyright © 2013 John Wiley & Sons, Ltd. We propose a well‐balanced stable generalized Riemann problem scheme for the shallow water equations with irregular bottom topography based on moving triangular meshes. Numerical tests show the accuracy, efficiency and robust of the scheme. In order to stabilize the computations near equilibria, we use the Rankine‐Hugoniot condition to remove a singularity from the GRP solver. We develop a remapping onto the new mesh (after grid movement) based on equilibrium variables. This, together with already established techniques, guarantees the well‐balancing. Numerical tests show the accuracy, efficiency, and robustness of the GRP moving mesh method: lake at rest solutions are preserved even when the underlying mesh is moving (e.g., mesh point are moved to regions of steep gradients) and various comparisons with fixed coarse and fine meshes demonstrate high resolution at relatively low cost.