Adaptive Mesh Refinement for Dam-Break Models using the Shallow Water Equations

Abstract

The 2D shallow water equations are a common tool for the simulation of free surface fluid dynamics in civil engineering. However, the nonlinear structures of the equations' straightforward implementations lead to numerical problems, such as spurious oscillations and unphysical diffusion. Therefore, this research compared several strategies to overcome these problems, using various finite element formulations and combinations of stabilization methods and mesh options. The accuracy and performance of numerous approaches are examined on models of dam-break in one and two space dimensions. The analytical solution checks the numerical, derived shock wave heights and velocities for the 1D classical benchmark. The result showed that streamlined diffusion and shock capturing stabilization deal with the classical problems of spurious oscillations and numerical diffusion but still indicate similar problems locally in the vicinity of steep fronts and shock waves when used on fixed meshes. As adaptive meshing is the most promising method to deal with such situations, several concerned options are examined in detail. It is important to fine-tune the method to the model's needs, i.e. to adapt the maximum number of mesh refinements, the indicator functions, and the starting mesh. The use of adaptive meshing techniques leads to accurate solutions for the usual parameter range in 1D and 2D, requiring less computational resources than simulations on fixed meshes. Meanwhile, meshing reduces the model size of the 2D dam break model adaptive by almost one order of magnitude and the execution time by a factor of 20.