10 citations · 40 across the 31 of their papers we have counts for
28 papers · 1 filter
Exact Total Variation Minimizers as Non-Oscillatory Limiters in High-Order Methods for Conservation Laws
Gabriel P. Langlois, Jerome Darbon, Rongjie Lai +2
High-order numerical methods for conservation laws can generate spurious oscillations near discontinuities. We propose to suppress these oscillations by post-processing the numeric…
Nonnegative Low-Rank Matrix Correction under an Orthogonality Constraint in Conservative Vlasov Simulations
Yue Wu, Stephen Becker, Jingmei Qiu +1
In low-rank numerical methods for Vlasov dynamics, the SVD-type truncation procedure may introduce negative entries into the numerical solution. Such negative values are unphysical…
A Simple GPU-Accelerated Solver for the Schrödinger Operator with Applications to Ground States and Hamiltonian Simulation
Xinyu Liu, Xiangxiong Zhang
We extend the tensor-product direct solver from the Laplacian to the Schrödinger operator . When the potential is separable, the operator is inverted or expo…
Efficient Admissible Set Projection in Optimization-based Invariant-Domain-Preserving Limiters for Ideal MHD
Chen Liu, Chi-Wang Shu, Xiangxiong Zhang
Preserving the admissible set of the ideal magnetohydrodynamics (MHD) equations is important not only for producing physically meaningful numerical solutions, but more importantly…
Computing the Gross-Pitaevskii Ground State via Wasserstein Gradient Flow in Diffeomorphism Space
Xiangxiong Zhang, Haomin Zhou
We compute the ground state of the Gross--Pitaevskii equation (GPE) via Wasserstein gradient descent in diffeomorphism space. We represent the density as the push-forwa…
High Order Numerical Methods Preserving Invariant Domain for Hyperbolic and Related Systems
Kailiang Wu, Xiangxiong Zhang, Chi-Wang Shu
Admissible states in hyperbolic systems and related equations often form a convex invariant domain. Numerical violations of this domain can lead to loss of hyperbolicity, resulting…
Fully discretized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem
Ziang Chen, Jianfeng Lu, Yulong Lu +1
This paper studies the numerical approximation of the ground state of the Gross-Pitaevskii (GP) eigenvalue problem with a fully discretized Sobolev gradient flow induced by the $H^…
An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier-Stokes equations
Chen Liu, Gregery T. Buzzard, Xiangxiong Zhang
We consider an optimization based limiter for enforcing positivity of internal energy in a semi-implicit scheme for solving gas dynamics equations. With Strang splitting, the compr…
On enforcing non-negativity in polynomial approximations in high dimensions
Yuan Chen, Dongbin Xiu, Xiangxiong Zhang
Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negat…
A high order accurate bound-preserving compact finite difference scheme for two-dimensional incompressible flow
Hao Li, Xiangxiong Zhang
For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving (SSP) temporal d…
On the monotonicity of spectral element method for Laplacian on quasi-uniform rectangular meshes
Logan J. Cross, Xiangxiong Zhang
The monotonicity of discrete Laplacian implies discrete maximum principle, which in general does not hold for high order schemes. The spectral element method has been proven…
A monotone finite element method for anisotropic elliptic equations
Hao Li, Xiangxiong Zhang
We construct a monotone continuous finite element method on the uniform mesh for the anisotropic diffusion problem with a diagonally dominant diffusion coefficient matrix. Th…
A simple GPU implementation of spectral-element methods for solving 3D Poisson type equations on rectangular domains and its applications
Xinyu Liu, Jie Shen, Xiangxiong Zhang
It is well known since 1960s that by exploring the tensor product structure of the discrete Laplacian on Cartesian meshes, one can develop a simple direct Poisson solver with an $\…
Riemannian Langevin Monte Carlo schemes for sampling PSD matrices with fixed rank
Tianmin Yu, Shixin Zheng, Jianfeng Lu +2
This paper introduces two explicit schemes to sample matrices from Gibbs distributions on , the manifold of real positive semi-definite (PSD) matrices of size $…
A simple and efficient convex optimization based bound-preserving high order accurate limiter for Cahn-Hilliard-Navier-Stokes system
Chen Liu, Beatrice Riviere, Jie Shen +1
For time-dependent PDEs, the numerical schemes can be rendered bound-preserving without losing conservation and accuracy, by a post processing procedure of solving a constrained mi…
A positivity-preserving implicit-explicit scheme with high order polynomial basis for compressible Navier-Stokes equations
Chen Liu, Xiangxiong Zhang
In this paper, we are interested in constructing a scheme solving compressible Navier--Stokes equations, with desired properties including high order spatial accuracy, conservation…
On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: a Riemannian optimization interpretation
Shixin Zheng, Haizhao Yang, Xiangxiong Zhang
In many applications, it is desired to obtain extreme eigenvalues and eigenvectors of large Hermitian matrices by efficient and compact algorithms. In particular, orthogonalization…
On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem
Ziang Chen, Jianfeng Lu, Yulong Lu +1
We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross-Pitaevskii eigenvalue problem. They are constructed as the gradient flows of th…
Structure preserving schemes for Fokker-Planck equations of irreversible processes
Chen Liu, Yuan Gao, Xiangxiong Zhang
In this paper, we introduce second order and fourth order space discretization via finite difference implementation of the finite element method for solving Fokker-Planck equations…
Discrete Maximum principle of a high order finite difference scheme for a generalized Allen-Cahn equation
Jie Shen, Xiangxiong Zhang
We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order…
Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations
Jingwei Hu, Xiangxiong Zhang
In this work, we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the li…
Accuracy of spectral element method for wave, parabolic and Schrödinger equations
Hao Li, Daniel Appelö, Xiangxiong Zhang
The spectral element method constructed by the () continuous finite element method with -point Gauss-Lobatto quadrature on rectangular meshes is a popular hig…
Monotonicity of spectral element method for discrete Laplacian
Logan J. Cross, Xiangxiong Zhang
The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schem…
On the monotonicity and discrete maximum principle of the finite difference implementation of - finite element method
Hao Li, Xiangxiong Zhang
We show that the fourth order accurate finite difference implementation of continuous finite element method with tensor product of quadratic polynomial basis is monotone thus satis…