Abstract
Non-uniform, dynamically adaptive meshes are a useful tool for reducing computational complexities for geophysical simulations that exhibit strongly localised features such as is the case for tsunami, hurricane or typhoon prediction. Using the example of a shallow water solver, this study explores a set of metrics as a tool to distinguish the performance of numerical methods using adaptively refined versus uniform meshes independent of computational architecture or implementation. These metrics allow us to quantify how a numerical simulation benefits from the use of adaptive mesh refinement. The type of meshes we are focusing on are adaptive triangular meshes that are non-uniform and structured. Refinement is controlled by physics-based indicators that capture relevant physical processes and determine the areas of mesh refinement and coarsening. The proposed performance metrics take into account a number of characteristics of numerical simulations such as numerical errors, spatial resolution, as well as computing time. Using a number of test cases we demonstrate that correlating different quantities offers insight into computational overhead, the distribution of numerical error across various mesh resolutions as well as the evolution of numerical error and run-time per degree of freedom.
Highlights
A large part of today’s research depends on computer simulations that are used to predict or simulate experiments that would be too difficult, costly, dangerous or, straight out, impossible to carry out in the real world
We ran a number of uniform and adaptive simulations with 6 ≤ λref ≤ 14 and a time step of Δt = 0.1s for the finest resolution h = 3.2 · 10−3m and larger time steps for coarser resolutions such that the CFL stability condition was fulfilled
We ran sets of simulations with uniform and adaptive mesh refinement with mesh parameters ranging from 8 ≤ λcrs, λref ≤ 17, θref = 0.05, and θcrs = 0.001 with a time step of Δt = 16s for h = 207.11m which corresponds to λref = 8 and smaller Δt for larger mesh levels in a way that the CFL condition is still fulfilled
Summary
A large part of today’s research depends on computer simulations that are used to predict or simulate experiments that would be too difficult, costly, dangerous or, straight out, impossible to carry out in the real world. The performance of an AMR method depends on refinement indicators ητi on elements τi that determine the areas to be refined These will require insight into the physical problem, are closely related to model sensitivities, and are needed for automated mesh manipulation as we will describe in more detail in Sect. Automatic mesh refinement has been shown to improve simulation accuracy [11] for idealised meteorological applications even when disjoint areas of high resolution are used It was shown in [18] that for advection problems, they can reduce diffusive and phase errors. This allows to discuss relative error distribution on the mesh levels and how AMR can influence the relationship of resolution to degrees of freedom depending on mesh parameters.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
