364 RWTH Publication No: 230476        2013        IGPM364.pdf
TITLE A Well-Balanced Stable GRP Scheme for Adaptive Unstructured Triangular Meshes
AUTHORS Feng Zhou, Guoxian Chen, Sebastian Noelle, Huai-Cheng Guo
ABSTRACT A well-balanced stable generalized Riemann problem (GRP) scheme under triangular meshes is proposed to solve shallow water equations (SWEs) involving irregular bottom topography, with the goal of the balance between flux gradients and bed slope source term, numerical stability, and high-performance computation. This scheme consists of two parts: an updated GRP evolution and adaptive mesh movement. The former is a second-order version Godunov method by evaluating the stability-preserving time-derivatives solutions along the exterior normal direction of the cell boundaries and by incorporating suitable central discretization of the bed source term. The latter is to move triangular meshes by iteratively solving Euler-Lagrange equations and remapping solutions on new meshes through geometrical conservative interpolations, which realizes still flat water property when the free surface is chosen as the interpolation variable. Three test cases were conducted to verify the performance of this updated scheme on the well-balanced stability-preserving property, predictive accuracy, spatial resolution, and computational efficiency. The results revealed two attractive features: (1) this scheme could preserve static flow by balancing bed slope and flux gradients through the application of unstructured triangular meshes with more stability through a modification of GRP time derivative calculation; (2) it could significantly improve the predictive accuracy and resolution of local features where gradients of flow variables are sharp, with a relatively lower computational cost.
KEYWORDS shallow water equations, generalized Riemann problem, adaptive unstructured meshes, well-balanced scheme, hydrodynamic process
DOI 10.1002/fld.3800
PUBLICATION International Journal for Numerical Methods in Fluids
Volume 73, Issue 3, pages 266–283, 30 September 2013