Abstract : Advection-diffusion PDEs represent a broad range of mathematical models in different fields of science and engineering, involving physical processes such as fluid flow, heat and mass transfer, diffusion, etc. Discretizations of advection-diffusion PDEs must be physically consistent, accurate and efficient on emerging computing architectures. The purpose of this minisymposium is to showcase recent developments in numerical methods that explore unconventional approaches to address these challenges. Methods such as residual distribution, flux-corrected transport, hybridized DG and FEM, SBP-SAT, machine learning enhanced methods, as well as methods for non-local extensions of classical advection-diffusion PDEs are of particular interest to this miniymposium.
Organizer(s) : Svetlana Tokareva, Nathaniel Morgan, Dmitri Kuzmin, Remi Abgrall
[03915] Dissipation-based WENO stabilization of high-order finite element methods for hyperbolic problems
Format : Online Talk on Zoom
Author(s) :
Dmitri Kuzmin (TU Dortmund University)
Joshua Vedral (TU Dortmund University)
Abstract : We propose a new kind of weighted essentially nonoscillatory (WENO) schemes to high-order finite element discretizations of hyperbolic conservation laws. In contrast to WENO-based limiters for DG approximations, our approach uses a reconstruction-based smoothness sensor to blend the numerical viscosity operators of high- and low-order stabilization terms. The so-defined hybrid approximation introduces low-order nonlinear diffusion in the vicinity of shocks, while preserving the high-order accuracy of the baseline discretization in regions where the exact solution is smooth. The underlying reconstruction procedure performs Hermite interpolation on stencils consisting of a mesh cell and its neighbors. The amount of numerical dissipation depends on the relative differences between partial derivatives of reconstructed candidate polynomials and those of the consistent finite element approximation. All derivatives are taken into account by the employed smoothness sensor. To assess the accuracy of our WENO scheme, we derive error estimates and perform numerical experiments. In particular, we prove that the consistency error of the nonlinear stabilization is of the order p+1/2, where p is the polynomial degree. For uniform meshes and smooth exact solutions, the experimentally observed rate of convergence is as high as p+1.
[04012] A Residual Distribution Approach to Isotropic Wave Kinetic Equations
Format : Talk at Waseda University
Author(s) :
Steven Walton (Los Alamos National Laboratory)
Abstract : We present a Petrov-Galerkin Residual Distribution (PG-RD) approach to solve isotropic wave kinetic equations (WKEs). The RD method is well-known for its ability to accurately solve advection dominated flow problems due to its intrinsic multi-dimensional upwinding property. While WKEs are nonlinear and non-local integro-differential equations, very different from the historical applications of RD methods, the energy flux of the equations we consider is local. We show that the PG-RD method derived generalizes a finite volume scheme given in a previous work. Analysis of the convergence properties of the method is also provided. The method is verified against established theoretical work for isotropic WKEs.
[04019] High order Flux Reconstruction schemes for turbulent flows and spectral analysis
Format : Talk at Waseda University
Author(s) :
ROMARIC SIMO TAMOU (IFPEN)
JULIEN BOHBOT (IFPEN)
JULIEN COATLÉVEN (IFPEN)
VINCENT PERRIER (INRIA)
QUANG HUY TRAN (IFPEN)
Abstract : This study focuses on evaluating Flux Reconstruction schemes for turbulent flows. For these schemes, we perform new analyses of their dissipation and dispersion properties, and we find consistent results with the classical analysis. Ultimately, we evaluate the effect of the high order and correction functions on the DNS of Taylor-Green vortex. This work provides valuable insights into the performance of FR schemes for turbulent flows and presents a promising new approach for analyzing their stability.
[04429] The Cartesian Grid Active Flux Method with Adaptive Mesh Refinement
Format : Talk at Waseda University
Author(s) :
Erik Chudzik
Christiane Helzel (Heinrich-Heine Universität)
Abstract : We present the first implementation of the Active Flux method on Cartesian grids with adaptive mesh refinement: A new finite volume method for hyperbolic conservation laws, that was introduced by Eymann and Roe, which uses a continuous, piecewise quadratic reconstruction and Simpson’s rule to compute numerical fluxes. Point values at grid cell interfaces together with cell averages are used to compute the reconstruction. The resulting method is third order accurate and has a compact stencil in space and time.
[04883] Optimization-based, property-preserving algorithm for passive tracer transport
Format : Talk at Waseda University
Author(s) :
Pavel Bochev (Sandia National Laboratories)
Kara Peterson (Sandia National Laboratories)
Denis Ridzal (Sanda National Laboratories)
Abstract : We present a new optimization-based property-preserving algorithm for passive tracer transport. The algorithm utilizes a semi-Lagrangian approach based on incremental remapping of the mass and the total tracer. However, unlike traditional semi-Lagrangian schemes, which remap the density and the tracer mixing ratio through monotone reconstruction or flux correction, we utilize an optimization-based remapping that enforces conservation and local bounds as optimization constraints.
In so doing we separate accuracy considerations from preservation of physical properties to obtain a conservative, second-order accurate transport scheme that also has a notion of optimality. Moreover, we prove that the optimization-based algorithm preserves linear relationships between tracer mixing ratios. We illustrate the properties of the new algorithm using a series of standard tracer transport test problems in a plane and on a sphere.
[04980] Multi-material ALE remap: interface sharpening in a matrix-free computation
Format : Talk at Waseda University
Author(s) :
Vladimir Z Tomov (Lawrence Livermore National Lab)
Tzanio Kolev (Lawrence Livermore National Laboratory)
Robert Rieben (Lawrence Livermore National Lab)
Arturo Vargas (Lawrence Livermore National Lab)
Abstract : We propose a new method for remap of material volume fractions, densities, and specific internal energies in the context of compressible ALE hydrodynamics. The remap is based on advection in pseudotime. As the volume fraction method can diffuse materials over many mesh elements, we introduce a sharpening modification on PDE level. We explain the effects of the modification and how it produces results that are still conservative and bounded. The latter involves FCT-type methods. The second major contribution, next to sharpening, is that all remap methods avoid assembly of global matrices. This avoids data motion and provides higher computational efficiency.
Performed under the auspices of the U.S. Department of Energy under Contract DE-AC52-07NA27344 (LLNL-ABS-847808)
[05104] Robust second-order approximation of the compressible Euler Equations with an arbitrary equation of state
Format : Talk at Waseda University
Author(s) :
Eric Joseph Tovar (Los Alamos National Laboratory)
Bennett Clayton (Texas A&M University)
Jean-Luc Guermond (Texas A&M University)
Matthias Maier (Texas A&M University)
Bojan Popov (Texas A&M University)
Abstract : This work is concerned with constructing a robust, high-order approximation of the compressible Euler equations for gas dynamics supplemented with an arbitrary or tabulated equation of state. In particular, we show how to construct a high-order graph-viscosity coefficient using an interpolated entropy pair useful when the equation of state is given by tabulated experimental data. Similarly, we construct an entropy surrogate functional that is used in a convex limiting technique that preserves the invariant domain of the system. Finally, the numerical method is then verified with analytical solutions and then validated with several benchmarks seen in the literature and laboratory experiments.
[05153] Heuristic Topological Estimation of Reduced Order Model Basis Functions from PDE Solution Snapshots
Format : Talk at Waseda University
Author(s) :
Candace Pauline Diaz (Sandia National Laboratories)
Pavel Bochev (Sandia National Laboratories)
Denis Ridzal (Sandia National Laboratories)
Abstract : Proper Orthogonal Decomposition (POD) is a common approach to obtain reduced basis sets for projection-based model reduction. For some classes of problems, such as hyperbolic Partial Differential Equations (PDEs), POD does not always achieve reasonable order reduction due to the lack of exponential decay of the leading singular values. In this work we investigate persistent homology as an alternative approach for deriving reduced order basis functions that may be particularly parsimonious for such PDEs.
