Adaptive third order Adams-Bashforth time integration for extended Boussinesq equations

Abstract

We develop the third-order adaptive Adams-Bashforth time integration and the second-order finite difference equation for variable time steps. We incorporate these schemes in the Celeris Advent software to discretize and solve the 2D extended Boussinesq equations. This software uses a hybrid finite volume – finite difference scheme and leverages the GPU to solve the equations faster than real-time while concurrently visualizing them. The newly added adaptive scheme significantly improves the robustness of the model while providing faster computational performance. We simulate several benchmarks using the adaptive time stepping scheme of Celeris Advent and demonstrate the capability of the software in modeling wave-breaking, wave runup, irregular waves, and rip currents. Program SummaryProgram title: Celeris Advent (v.1.3.4)CPC Library link to program files:https://doi.org/10.17632/pwsjdsgz89.1Licensing provisions: GNU General Public License 3Programming language: C++, HLSLNature of problem: Celeris Advent started a new paradigm in nearshore wave simulations and enabled researchers and engineers to run a Boussinesq-type model, faster than real-time and in an interactive environment. For simplicity, we assumed a fixed time step in our first implementation of Celeris Advent. This fixed time step often needs to be chosen conservatively such that the model can resolve the most extreme cases during the experiment. In practical simulations, such as simulating coastal fields, the superposition of boundary and initial conditions may cause rare but extreme conditions, requiring a very small time step that is too conservative during most of the simulation.Solution method: We developed adaptive third order Adams-Bashforth time integration to let Celeris Advent solve the extended Boussinesq equations with a variable time step, allowing it to decrease the time step only when necessary. The adaptive equations are presented in a generic format and therefore can be used for solving other equations as well.Additional comments including restrictions and unusual features: The new version of the Celeris Advent with the adaptive time integration runs ∼3 times faster for the standard conical island benchmark, allowing Celeris Advent simulate this benchmark on a 200×200 grid an order of magnitude faster than real-time on a consumer-level gaming laptop. For a field simulation benchmark, with rare but extreme events, the new version runs ∼25 times faster.