Jie DuYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, ChinaYang YangDepartment of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA
Numerical Analysis and Scientific Computingmathscidoc:2205.25022
Communications on Applied Mathematics and Computation, 2021.4
In this paper, we apply high-order finite difference (FD) schemes for multispecies and multireaction detonations (MMD). In MMD, the density and pressure are positive and the mass fraction of the ith species in the chemical reaction, say zi, is between 0 and 1, with ∑z_i=1. Due to the lack of maximum-principle, most of the previous bound-preserving technique cannot be applied directly. To preserve those bounds, we will use the positivity-preserving technique to all the z′is and enforce ∑z_i=1 by constructing conservative schemes, thanks to conservative time integrations and consistent numerical fluxes in the system. Moreover, detonation is an extreme singular mode of flame propagation in premixed gas, and the model contains a significant stiff source. It is well known that for hyperbolic equations with stiff source, the transition points in the numerical approximations near the shocks may trigger spurious shock speed, leading to wrong shock position. Intuitively, the high-order weighted essentially non-oscillatory (WENO) scheme, which can suppress oscillations near the discontinuities, would be a good choice for spatial discretization. However, with the nonlinear weights, the numerical fluxes are no longer “consistent”, leading to nonconservative numerical schemes and the bound-preserving technique does not work. Numerical experiments demonstrate that, without further numerical techniques such as subcell resolutions, the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.
Jie DuYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, ChinaEric ChungDepartment of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR, ChinaYang YangDepartment of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA
Numerical Analysis and Scientific Computingmathscidoc:2205.25021
Communications on Applied Mathematics and Computation, 4, 353-379, 2022.4
In this paper, we study the classical Allen-Cahn equations and investigate the maximum-principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen-Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.
Jie DuYau Mathematical Sciences Center, Tsinghua University, Beijing, ChinaEric T. ChungDepartment of Mathematics, The Chinese University of Hong Kong, Sha Tin, Hong Kong SARMing Fai LamDepartment of Mathematics, The Chinese University of Hong Kong, Sha Tin, Hong Kong SARXiao-Ping WangDepartment of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR
Numerical Analysis and Scientific Computingmathscidoc:2205.25020
Journal of Scientific Computing, 76, 1547–1577, 2018.3
In this paper, we present a discontinuous Galerkin method with staggered hybridization to discretize a class of nonlinear Stokes equations in two dimensions. The utilization of staggered hybridization is new and this approach combines the features of traditional hybridization method and staggered discontinuous Galerkin method. The main idea of our method is to use hybrid variables to impose the staggered continuity conditions instead of enforcing them in the approximation space. Therefore, our method enjoys some distinctive advantages, including mass conservation, optimal convergence and preservation of symmetry of the stress tensor. We will also show that, one can obtain superconvergent and strongly divergence-free velocity by applying a local postprocessing technique on the approximate solution. We will analyze the stability and derive a priori error estimates of the proposed scheme. The resulting nonlinear system is solved by using the Newton’s method, and some numerical results will be demonstrated to confirm the theoretical rates of convergence and superconvergence.
Eric T. ChungDepartment of Mathematics, The Chinese University of Hong Kong, Hong Kong, SAR, People’s Republic of ChinaJie DuDepartment of Mathematics, The Chinese University of Hong Kong, Hong Kong, SAR, People’s Republic of ChinaMan Chun YuenDepartment of Mathematics, The Chinese University of Hong Kong, Hong Kong, SAR, People’s Republic of China
Numerical Analysis and Scientific Computingmathscidoc:2205.25019
Journal of Scientific Computing, 70, 766-792, 2016.8
Staggered grid techniques are attractive ideas for flow problems due to their more enhanced conservation properties. Recently, a staggered discontinuous Galerkin method is developed for the Stokes system. This method has several distinctive advantages, namely high order optimal convergence as well as local and global conservation properties. In addition, a local postprocessing technique is developed, and the postprocessed velocity is superconvergent and pointwisely divergence-free. Thus, the staggered discontinuous Galerkin method provides a convincing alternative to existing schemes. For problems with corner singularities and flows in porous media, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, we will derive a computable error indicator for the staggered discontinuous Galerkin method and prove that this indicator is both efficient and reliable. Moreover, we will present some numerical results with corner singularities and flows in porous media to show that the proposed error indicator gives a good performance.
Xin LiuDepartment of Mathematics, Southern University of Science and Technology, Shenzhen, 518055,China, and Numerical Environmental Prediction Section, Canadian Meteorological Centre, Environ-ment and Climate Change Canada, Dorval, QC, H9P 1J3, CanadaXin ChenDepartment of Mathematics, Harbin Institute of Technology, Harbin, 150001, China, and De-partment of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, ChinaShi JinSchool of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, Shanghai Jiao TongUniversity, Shanghai, 200240, ChinaAlexander KurganovDepartment of Mathematics and SUSTech International Center for Mathematics, Southern Uni-versity of Science and Technology, Shenzhen, 518055, ChinaTong WuMathematics Department, Tulane University, New Orleans, LA 70118Hui YuYau Mathematical Science Center, Tsinghua University, Beijing, 100084, China
Numerical Analysis and Scientific Computingmathscidoc:2205.25018
SIAM Journal on Scientific Computing, 42, (4), A2206-A2229, 2020.7
We develop a new moving-water equilibria preserving numerical scheme for the Saint-Venant system. The new scheme is designed in two major steps. First, the geometric source term is incorporated into the discharge flux, which results in a hyperbolic system with a global flux. Second, the discharge equation is relaxed so that the nonlinearity is moved into the stiff right-hand side of the added auxiliary equation. The main advantages of the new scheme are that (i) no special treatment of the geometric source term is required, and (ii) no nonlinear (cubic) equations should be solved to obtain the point values of the water depth out of the reconstructed equilibrium variables, as it must be done in the existing alternative methods. We also develop a hybrid numerical flux, which helps to handle various flow regimes in a stable manner. Several numerical experiments are performed to verify that the proposed scheme is capable of exactly preserving general moving-water steady states and accurately capturing their small perturbations.