Quadrature-free discontinuous Galerkin method with code generation features for shallow water equations on automatically generated block-structured meshes

p-adaptive discontinuous Galerkin method for the shallow water equations with a parameter-free error indicator

“…The simulations were performed with the ExaStencils code generation framework (Lengauer et al 2020), which is based on the domain-specific language ExaSlang (Schmitt et al 2014b) and outputs an optimized C++ or CUDA code. Our work extends the framework using a Python frontend -called GHODDESS and available at https:// i10git.cs.fau.de/ocean/ghoddess-release -responsible for mapping the DG scheme to ExaSlang (Faghih-Naini et al 2020). Recently, much effort was put into improving the performance of our p-adaptive scheme via algorithmic optimizations and various hybridization strategies controlling the distribution of the compute kernels between CPUs and GPUs.…”

“…The presentation in this section closely follows (Faghih-Naini et al 2020) and is included here for completeness. We start with the classical 2D shallow water equations defined on some 2D domain Ω and given by…”

“…With the help of the auxiliary vector u calculated diagnostically from (5), this formulation, first presented in Faghih-Naini et al (2020), avoids fraction-type nonlinearities in the momentum equation ( 2) and is better suited for quadrature-free integration. This work uses the following types of boundary conditions for the SWE:…”

“…sensitivity) parameters, and is integrated into the vertex-based slope limiter (Aizinger 2011;Kuzmin 2010;Aizinger et al 2017;Hajduk et al 2018Hajduk et al , 2019 further reducing the total computational cost of the scheme. In addition, our DG discretization based on modal hierarchical bases is highly suited for p-adaptive schemes -even more so since it relies on a quadrature-free formulation introduced in Faghih- Naini et al (2020). Analytic evaluation of all element and edge integrals completely avoids the main overhead of varying-order approximation spaces: The necessity to use the most accurate (and thus the most expensive) quadrature rule or to maintain several quadrature rules of different orders.…”

Enriched Galerkin method for the shallow-water equations

“…Several interesting avenues of future research concerning the development of an EG solver for the SWE present themselves at this time. Thus one could try the quadraturefree methodology to improve the computational efficiency of the method-similarly to our recent work for the DG SWE solver in Faghih-Naini et al (2020). Formulating the EG method for the SWE as a time-implicit or a semi-implicit scheme-especially in a combination with an efficient linear solver such as the hierarchical scale separation method either directly (Aizinger et al 2015) or in a hybridized setting (Schütz and Aizinger 2017) appears to be particularly attractive due to similarities with the DG method, on one hand, and a substantially smaller system size, on the other.…”