A parallel wavelet adaptive WENO scheme for 2D conservation laws

Abstract

Purpose - The current work aims to present a parallel code using the open multi-processing (OpenMP) programming model for an adaptive multi-resolution high-order finite difference scheme for solving 2D conservation laws, comparing efficiencies obtained with a previousmessage passing interface formulation for the same serial scheme and considering the same type of 2D formulations laws. Design/methodology/approach - The serial version of the code is naturally suitable for parallelization because the spatial operator formulation is based on a splitting scheme per direction for which the flux components are numerically computed by a Lax-Friedrichs factorization independently for each row or column. High-order approximations for numerical fluxes are computed by the third-order essentially non-oscillatory (ENO) and fifth-order weighted essentially non-oscillatory (WENO) interpolation schemes, assuming sparse grids in each direction. The grid adaptivity is obtained by a cubic interpolating wavelet transform applied in each space dimension, associated to a threshold operator. Time is evolved by a third order TVD Runge-Kutta method. Findings - The parallel formulation is implemented automatically at compiling time by the OpenMP library routines, being virtually transparent to the programmer. This over simplifies any concerns about managing and/ or updating the adaptive grid when compared to what is necessary to be done when other parallel approaches are considered. Numerical simulations results and the large speedups obtained for the Euler equations in gas dynamics highlight the efficiency of the OpenMP approach. Research limitations/implications - The resulting speedups reflect the effectiveness of the OpenMP approach but are, to a large extension, limited by the hardware used (2 E5-2620 Intel Xeon processors, 6 cores, 2 threads/core, hyper-threading enabled). As the demand for OpenMP threads increases, the code starts to make explicit use of the second logical thread available in each E5-2620 processor core and efficiency drops. The speedup peak is reached near the possible maximum (24) at about 22, 23 threads. This peak reflects the hardware configuration and the true software limit should be located way beyond this value. Practical implications - So far no attempts have been made to parallelize other possible code segments (for instance, the ENO|-WENO-TVD code lines that process the different data components which could potentially push the speed up limit to higher values even further. The fact that the speedup peak is located close to the present hardware limit reflects the scalability properties of the OpenMP programming and of the splitting scheme as well. Consequently, it is likely that the speedup peak with the OpenMP approach for this kind of problem formulation will be close to the physical (and/or logical) limit of the hardware used. Social implications - This work is the result of a successful collaboration among researchers from two different institutions, one internationally well-known and with a long-term ...

Solving a mixture model of two-phase flow with velocity non-equilibrium using WENO wavelet methods

“…In Schmidt et al ’s work (2017), a study was presented analyzing the error depending on CFL and grid size. Both the WENO-JS and the ENO interpolations for the Lax–Friedrichs flux splitting formulation without wavelet analysis were computed for a series of different values of CFL, from 0.1 to 0.9 and for grids of 2 8 , 2 9 and 2 10 points, assuming as benchmark hyperbolic conservation laws problems.…”

“…In the context of adaptive wavelet methods, one contribution with respect to the stability analysis of the multiresolution scheme was given by Dubos and Kevlahan (2013) for solving shallow-water equations on staggered grids. For the current work, the numerical evidences of the stability of the considered numerical scheme have been provided in the study presented by Schmidt et al (2017), in which a wide range of acceptable CFL values has been evinced for initial value problems for hyperbolic conservation laws.…”

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

“…[15][16][17] Finally, wavelet-optimized methods include algorithms based on classical discretizations (eg, by finite differences or finite volumes) but use wavelet analysis either to define adaptive meshes or to speed up linear algebra. 19,[26][27][28][29][30] In our investigation, we utilize the latter approach. We note here that multiresolution analysis is utilized for the latter 2 approaches, but wavelet-optimized methods actually adapt grids dynamically from information obtained through this analysis.…”

“…In this regard, we may divide wavelet methods into 3 main classes as follows: pure wavelet, multiresolution, and wavelet optimized methods. 19,[26][27][28][29][30] In our investigation, we utilize the latter approach. Furthermore, adaptive wavelet Galerkin methods 24 are used to solve problems in wavelet coefficient space, whereas adaptive wavelet collocation methods 25 are used to solve problems in physical space on an adaptive computational grid.…”

An adaptive space-time shock capturing method with high order wavelet bases for the system of shallow water equations

“…Moreover, wavelet bases have successfully been used in finite difference (Farge and Schneider, 2015) methods in particular for flow simulations (Schneider and Vasilyev, 2010). The capability of wavelets in singularity detection has also been used in the context of finite volume and dG methods for hyperbolic problems (Domingues et al , 2009; Hovhannisyan et al , 2014; Jiwari et al , 2017; Schmidt et al , 2017) as well as boundary element method (Eppler and Harbrecht, 2005). This feature of wavelet bases has also been used to solve optimal control problems (Razzaghi and Yousefi, 2002), eigenvalue problems (Mollet et al , 2013) and ordinary differential equations (Homei et al , 2014) adaptively.…”