We propose a change of variable approach and discontinuity capturing methods to ensure physical constraints for advection-reaction equations discretized by the finite element method. This change of variable confines the concentration below an upper bound in a very natural way. For the non-negativity constraint, we propose to use a discontinuity capturing method defined on the reference element that is combined with an anisotropic crosswind-dissipation operator. This discontinuity capturing cannot completely eliminate negative values but effectively minimizes their occurrence. The proposed methods are applied to different biophysical models and show a good agreement with experimental results for the FDA benchmark blood pump for a physiological red blood cell pore formation model.
A categorical embedding discontinuity-capturing shallow neural network for anisotropic elliptic interface problems
Department of Mathematics, National Central University（数学系，国立中央大学）；Department of Applied Mathematics, National Yang Ming Chiao Tung University（应用数学系，国立阳明交通大学）；Department of Applied Mathematics, National University of Kaohsiung（应用数学系，国立高雄大学）
We study in this paper new developments of the Lagrange-Galerkin method for the advection equation. In the first part of the article we present a new improved error estimate of the conventional Lagrange-Galerkin method. In the second part, we introduce a new local projection stabilized Lagrange-Galerkin method, whereas in the third part we introduce and analyze a discontinuity-capturing Lagrange-Galerkin method. Also, attention has been paid to the influence of the quadrature rules on the stability and accuracy of the methods via numerical experiments.
The nonlinear convection terms in the governing equations of compressible fluid flows are hyperbolic in nature and are nontrivial for modelling and numerical simulation. Many numerical methods have been developed in the last few decades for this purpose and are typically based on Riemann solvers, which are strongly dependent on the underlying eigen-structure of the governing equations. Objective of the present work is to develop simple algorithms which are not dependent on the eigen-structure and yet can tackle easily the hyperbolic parts. Central schemes with smart diffusion mechanisms are ap…
Stable and accurate interface capturing advection schemes
CMLA, ENS Cachan, CNRS, Université Paris-Saclay（CMLA，巴黎高等师范学校卡尚分校，法国国家科学研究中心，巴黎萨克雷大学）；Maison de la simulation, USR 3441, CEA Saclay（模拟之家，USR 3441，CEA萨克莱）；CEA DAM DIF；CEA Saclay, DEN（CEA萨克莱，DEN）；CGG
A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems
National Central University（国立中央大学）；National Center for Theoretical Sciences, National Taiwan University（国立台湾大学国家理论科学研究中心）；National Yang Ming Chiao Tung University（国立阳明交通大学）
We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds a sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu prob…
A main disadvantage of many high-order methods for hyperbolic conservation laws lies in the famous Gibbs-Wilbraham phenomenon, once discontinuities appear in the solution. Due to the Gibbs-Wilbraham phenomenon, the numerical approximation will be polluted by spurious oscillations, which produce unphysical numerical solutions and might finally blow up the computation. In this work, we propose a new shock capturing procedure to stabilise high-order spectral element approximations. The procedure consists of going over from the original (polluted) approximation to a convex combination of the origi…
An interior penalty DG method with correct and minimal averages, jumps and penalties for the miscible displacement problem of nonnegative characteristic form, and SUPG-type error estimates under low regularity, dominating Darcy velocity
This work is devoted to the development of an efficient and robust technique for accurate capturing of the electric field in multi-material problems. The formulation is based on the finite element method enriched by the introduction of hat-type shape function within the elements crossed by the material interface. The peculiar feature of the proposed method consists in the direct employment of the hat-function that requires solely one additional degree of freedom per cut element for capturing the discontinuity in the electric potential gradient and, thus, the electric field. This additional deg…
This paper develops the structure-preserving finite volume weighted essentially non-oscillatory (WENO) hybrid schemes for the shallow water equations under the arbitrary Lagrangian-Eulerian (ALE) framework, dubbed as ALE-WENO schemes. The WENO hybrid reconstruction is adopted on moving meshes, which distinguishes the smooth, non-smooth, and transition stencils by a simple smoothness detector. To maintain the positivity preserving and the well-balanced properties of the ALE-WENO schemes, we adapt the positivity preserving limiter and the well-balanced approaches on static meshes to moving meshe…
Structure-Preserving Digital Twins via Conditional Neural Whitney Forms
Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA（应用力学，宾夕法尼亚大学，费城，PA，美国）；Sandia National Laboratories, Albuquerque, NM, USA（桑迪亚国家实验室，阿尔伯克基，NM，美国）
A high order finite difference method is proposed for unstructured meshes to simulate compressible inviscid/viscous flows with/without discontinuities. In this method, based on the strong form equation, the divergence of the flux on each vertex is computed directly from fluxes nearby by means of high order least-square. In order to capture discontinuities, numerical flux of high order accuracy is calculated on each edge and serves as supporting data of the least-square computation of the divergence. The high accuracy of the numerical flux depends on the high order WENO interpolation on each ed…
