Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2207.13736

A mass conservative Eulerian-Lagrangian Runge-Kutta discontinuous Galerkin method for wave equations with large time stepping

Xue Hong·Jing-Mei Qiu
View original source ↗Read source document ↗Export metadata ↓

Abstract

We propose an Eulerian-Lagrangian (EL) Runge-Kutta (RK) discontinuous Galerkin (DG) method for wave equations. The method is designed based on the ELDG method for transport problems [J. Comput. Phy. 446: 110632, 2021.], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping sizes with stability. The wave equation can be written as a first order hyperbolic system. Considering each characteristic family, a straightforward application of ELDG will be to transform to the characteristic variables, evolve them on associated characteristic related space-time regions, and transform them back to the original variables. However, the mass conservation could not be guaranteed in a general setting. In this paper, we formulate a mass conservative semi-discrete ELDG method by decomposing each variable into two parts, each of them associated with a different characteristic family. As a result, four different quantities are evolved in EL fashion and recombined to update the solution. The fully discrete scheme is formulated by using method-of-lines RK methods, with intermediate RK solutions updated on the background mesh. Numerical results on 1D and 2D wave equations are presented to demonstrate the performance of the proposed ELDG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and mass conservative property.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xue Hong, Jing-Mei Qiu. 2022-07-27. A mass conservative Eulerian-Lagrangian Runge-Kutta discontinuous Galerkin method for wave equations with large time stepping. https://arxiv.org/abs/2207.13736

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$L^{p}$-convergence of Kantorovich-type Max-Min Neural Network Operators

In this work, we study the Kantorovich variant of max-min neural network operators, in which the operator kernel is defined in terms of sigmoidal functions. Our main aim is to demonstrate the $L^{p}$-convergence of these nonlinear operators for $1\leq p<\infty$, which makes it possible to obtain approximation results for functions that are not necessarily continuous. In addition, we will derive quantitative estimates for the rate of approximation in the $L^{p}$-norm. We will provide some explicit examples, studying the approximation of discontinuous functions with the max-min operator, and varying additionally the underlying sigmoidal function of the kernel. Further, we numerically compare the $L^{p}$-approximation error with the respective error of the Kantorovich variants of other popular neural network operators. As a final application, we show that the Kantorovich variant has advantages compared to the sampling variant of the max-min operator and Kantorovich variant of the max-product operator when it comes to approximate noisy functions as for instance biomedical ECG signals.

math.NA↗
arXiv2026

Quotient geometry of tensor ring decomposition

Differential geometries derived from tensor decompositions have been extensively studied and provided the foundations for a variety of efficient numerical methods. Despite the practical success of the tensor ring (TR) decomposition, its intrinsic geometry remains less understood, primarily due to the underlying ring structure and the resulting nontrivial gauge invariance. We establish the quotient geometry and immersed-submanifold structure of TR decomposition by imposing full-rank conditions on all unfolding matrices of the core tensors and capturing the gauge invariance. The intrinsic ring structure of TR leads to an analysis that is substantially different from other tensor formats. Additionally, for the uniform TR decomposition, where all core tensors are identical and the manifold structure is known, we derive explicit parameterizations for the vertical and horizontal spaces, which enable Riemannian optimization. Numerical experiments validate the developed geometries via tensor ring completion tasks.

math.NA↗
arXiv2026

Boundary elements for clamped Kirchhoff--Love plates

We present a Galerkin boundary element method for clamped Kirchhoff--Love plates with piecewise smooth boundary. It is a direct method based on the representation formula and requires the inversion of the single-layer operator, an application of the double-layer operator to the Dirichlet data, and, in the presence of a vertical load, an application of the Dirichlet trace of the Newton potential to that load. We present trace approximation spaces of arbitrary order, required for both the Dirichlet data and the unknown Neumann trace. Our boundary element method is quasi-optimal with respect to the natural trace norm and achieves optimal convergence order under minimal regularity assumptions. We provide explicit representations of all three integral operators and discuss the implementation of the appearing integrals. Numerical experiments for smooth and non-smooth domains confirm predicted convergence rates.

math.NA↗