arXiv subjects

Weizhang Huang

Publications and source records attributed to Weizhang Huang.

At least 37 records · Page 2Linked to original sources

arXiv2020

A well-balanced positivity-preserving quasi-Lagrange moving mesh DG method for the shallow water equations

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of an update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.

math.NA↗

arXiv2020

An Introduction to MMPDElab

This article presents an introduction to MMPDElab, a package written in MATLAB for adaptive mesh movement and adaptive moving mesh P1 finite element solution of second-order partial different equations having continuous solutions in one, two, and three spatial dimensions. MMPDElab uses simplicial meshes.

math.NA↗

arXiv2020

High-order conservative positivity-preserving DG-interpolation for deforming meshes and application to moving mesh DG simulation of radiative transfer

Solution interpolation between deforming meshes is an important component for several applications in scientific computing, including indirect arbitrary-Lagrangian-Eulerian and rezoning moving mesh methods in numerical solution of partial differential equations. In this paper, a high-order, conservative, and positivity-preserving interpolation scheme is developed based on the discontinuous Galerkin solution of a linear time-dependent equation on deforming meshes. The scheme works for bounded but otherwise arbitrary mesh deformation from the old mesh to the new one. The cost and positivity preservation (with a linear scaling limiter) of the DG-interpolation are investigated. Numerical examples are presented to demonstrate the properties of the interpolation scheme. The DG-interpolation is applied to the rezoning moving mesh DG solution of the radiative transfer equation, an integro-differential equation modeling the conservation of photons and involving time, space, and angular variables. Numerical results obtained for examples in one and two spatial dimensions with various settings show that the resulting rezoning moving mesh DG method maintains the same convergence order as the standard DG method, is more efficient than the method with a fixed uniform mesh, and is able to preserve the positivity of the radiative intensity.

math.NA↗

arXiv2020

A high-order well-balanced positivity-preserving moving mesh DG method for the shallow water equations with non-flat bottom topography

A rezoning-type adaptive moving mesh discontinuous Galerkin method is proposed for the numerical solution of the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the simulation of perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance and positivity-preserving properties, strategies are discussed in the use of slope limiting, positivity-preservation limiting, and data transferring between meshes. Particularly, it is suggested that a DG-interpolation scheme be used for the interpolation of both the flow variables and bottom topography from the old mesh to the new one and after each application of the positivity-preservation limiting on the water depth, a high-order correction be made to the approximation of the bottom topography according to the modifications in the water depth. Moreover, mesh adaptation based on the equilibrium variable and water depth is shown to give more desirable results than that based on the commonly used entropy function. Numerical examples in one and two spatial dimensions are presented to demonstrate the well-balance and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state.

math.NA↗

arXiv2020

Domain Decomposition Parabolic Monge-Ampère Approach for Fast Generation of Adaptive Moving Meshes

A fast method is presented for adaptive moving mesh generation in multi-dimensions using a domain decomposition parabolic Monge-Ampère approach. The domain decomposition procedure employed here is non-iterative and involves splitting the computational domain into overlapping subdomains. An adaptive mesh on each subdomain is then computed as the image of the solution of the $L^2$ optimal mass transfer problem using a parabolic Monge-Ampère method. The domain decomposition approach allows straightforward implementation for the parallel computation of adaptive meshes which helps to reduce computational time significantly. Results are presented to show the numerical convergence of the domain decomposition solution to the single domain solution. Several numerical experiments are given to demonstrate the performance and efficiency of the proposed method. The numerical results indicate that the domain decomposition parabolic Monge-Ampère method is more efficient than the standard implementation of the parabolic Monge-Ampère method on the whole domain, in particular when computing adaptive meshes in three spatial dimensions.

math.NA↗

arXiv2020

Anisotropic mesh quality measures and adaptation for polygonal meshes

Anisotropic mesh quality measures and anisotropic mesh adaptation are studied for polygonal meshes. Three sets of alignment and equidistribution measures are developed, one based on least squares fitting, one based on generalized barycentric mapping, and the other based on singular value decomposition of edge matrices. Numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes under given metrics. Based on one of these sets of quality measures and using a moving mesh partial differential equation, an anisotropic adaptive polygonal mesh method is constructed for the numerical solution of second order elliptic equations. Numerical examples are presented to demonstrate the effectiveness of the method.

math.NA↗

arXiv2020

Permanent charge effects on ionic flow: a numerical study of flux ratios and their bifurcation

Ionic flow carries electrical signals for cells to communicate with each other. The permanent charge of an ion channel is a crucial protein structure for flow properties while boundary conditions play a role of the driving force. Their effects on flow properties have been analyzed via a quasi-one-dimensional Poisson-Nernst-Planck model for small and relatively large permanent charges. The analytical studies have led to the introduction of flux ratios that reflect permanent charge effects and have a universal property. The studies also show that the flux ratios have different behaviors for small and large permanent charges. However, the existing analytical techniques can reveal neither behaviors of flux ratios nor transitions between small and large permanent charges. In this work we present a numerical investigation on flux ratios to bridge between small and large permanent charges. Numerical results verify the analytical predictions for the two extremal regions. More significantly, emergence of non-trivial behaviors is detected as the permanent charge varies from small to large. In particular, saddle-node bifurcations of flux ratios are revealed, showing rich phenomena of permanent charge effects by the power of combining analytical and numerical techniques. An adaptive moving mesh finite element method is used in the numerical studies.

math.NA↗

arXiv2019

Conditioning of implicit Runge-Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes

The conditioning of implicit Runge-Kutta (RK) integration for linear finite element approximation of diffusion equations on general anisotropic meshes is investigated. Bounds are established for the condition number of the resulting linear system with and without diagonal preconditioning for the implicit Euler and general implicit RK methods. Two solution strategies are considered for the linear system resulting from general implicit RK integration: the simultaneous solution (the system is solved as a whole) and a successive solution which follows the commonly used implementation of implicit RK methods to first transform the system into smaller systems using the Jordan normal form of the RK matrix and then solve them successively. For the simultaneous solution in case of a positive semidefinite symmetric part of the RK coefficient matrix and for the successive solution it is shown that . If the smallest eigenvalue of the symmetric part of the RK coefficient matrix is negative and the simultaneous solution strategy is used, an upper bound on the time step is given so that the system matrix is positive definite. The obtained bounds for the condition number have explicit geometric interpretations and take the interplay between the diffusion matrix and the mesh geometry into full consideration. They show that there are three mesh-dependent factors that can affect the conditioning: the number of elements, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity with respect to the inverse of the diffusion matrix. They also reveal that the preconditioning using the diagonal of the system matrix, the mass matrix, or the lumped mass matrix can effectively eliminate the effects of the mesh nonuniformity measured in the Euclidean metric. Numerical examples are given.

math.NA↗

arXiv2019

Stability of explicit Runge-Kutta methods for high order finite element approximation of linear parabolic equations

We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.

math.NA↗

arXiv2019

A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauss--Seidel (GS) method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.

math.NA↗

arXiv2019

A surface moving mesh method based on equidistribution and alignment

A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed even when the surface only has a numerical representation. A selection of two- and three-dimensional examples are presented.

math.NA↗

arXiv2018

A quasi-Lagrangian moving mesh discontinuous Galerkin method for hyperbolic conservation laws

A moving mesh discontinuous Galerkin method is presented for the numerical solution of hyperbolic conservation laws. The method is a combination of the discontinuous Galerkin method and the mesh movement strategy which is based on the moving mesh partial differential equation approach and moves the mesh continuously in time and orderly in space. It discretizes hyperbolic conservation laws on moving meshes in the quasi-Lagrangian fashion with which the mesh movement is treated continuously and no interpolation is needed for physical variables from the old mesh to the new one. Two convection terms are induced by the mesh movement and their discretization is incorporated naturally in the DG formulation. Numerical results for a selection of one- and two-dimensional scalar and system conservation laws are presented. It is shown that the moving mesh DG method achieves the theoretically predicted order of convergence for problems with smooth solutions and is able to capture shocks and concentrate mesh points in non-smooth regions. Its advantage over uniform meshes and its insensitiveness to mesh smoothness are also demonstrated.

math.NA↗

arXiv2018

An adaptive moving mesh discontinuous Galerkin method for the radiative transfer equation

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, space and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are demonstrated in a selection of one- and two-dimensional numerical examples.

math.NA↗

arXiv2018

A Study on Phase-Field Models for Brittle Fracture

In the phase-field modeling of brittle fracture, anisotropic constitutive assumptions for the degradation of stored elastic energy due to fracture are crucial to preventing cracking in compression and obtaining physically sound numerical solutions. Three energy decomposition models, the spectral decomposition, the volumetric-deviatoric split, and an improved volumetric-deviatoric split, and their effects on the performance of the phase-field modeling are studied. Meanwhile, anisotropic degradation of stiffness may lead to a small energy remaining on crack surfaces, which violates crack boundary conditions and can cause unphysical crack openings and propagation. A simple yet effective treatment for this is proposed: define a critically damaged zone with a threshold parameter and then degrade both the active and passive energies in the zone. A dynamic mesh adaptation finite element method is employed for the numerical solution of the corresponding elasticity system. Four examples, including two benchmark ones, one with complex crack systems, and one based on an experimental setting, are considered. Numerical results show that the spectral decomposition and improved volumetric-deviatoric split models, together with the improvement treatment of crack boundary conditions, can lead to crack propagation results that are comparable with the existing computational and experimental results. It is also shown that the numerical results are not very sensitive to the parameter defining the critically damaged zone.

math.NA↗

arXiv2018

Moving Mesh simulation of contact sets in two dimensional models of elastic-electrostatic deflection problems

Numerical and analytical methods are developed for the investigation of contact sets in electrostatic-elastic deflections modeling micro-electro mechanical systems. The model for the membrane deflection is a fourth-order semi-linear partial differential equation and the contact events occur in this system as finite time singularities. Primary research interest is in the dependence of the contact set on model parameters and the geometry of the domain. An adaptive numerical strategy is developed based on a moving mesh partial differential equation to dynamically relocate a fixed number of mesh points to increase density where the solution has fine scale detail, particularly in the vicinity of forming singularities. To complement this computational tool, a singular perturbation analysis is used to develop a geometric theory for predicting the possible contact sets. The validity of these two approaches are demonstrated with a variety of test cases.

math.NA↗

arXiv2018

Moving mesh finite difference solution of non-equilibrium radiation diffusion equations

A moving mesh finite difference method based on the moving mesh partial differential equation is proposed for the numerical solution of the 2T model for multi-material, non-equilibrium radiation diffusion equations. The model involves nonlinear diffusion coefficients and its solutions stay positive for all time when they are positive initially. Nonlinear diffusion and preservation of solution positivity pose challenges in the numerical solution of the model. A coefficient-freezing predictor-corrector method is used for nonlinear diffusion while a cutoff strategy with a positive threshold is used to keep the solutions positive. Furthermore, a two-level moving mesh strategy and a sparse matrix solver are used to improve the efficiency of the computation. Numerical results for a selection of examples of multi-material non-equilibrium radiation diffusion show that the method is capable of capturing the profiles and local structures of Marshak waves with adequate mesh concentration. The obtained numerical solutions are in good agreement with those in the existing literature. Comparison studies are also made between uniform and adaptive moving meshes and between one-level and two-level moving meshes.

math.NA↗

arXiv2018

Conditioning of Finite Volume Element Method for Diffusion Problems with General Simplicial Meshes

The conditioning of the linear finite volume element discretization for general diffusion equations is studied on arbitrary simplicial meshes. The condition number is defined as the ratio of the maximal singular value of the stiffness matrix to the minimal eigenvalue of its symmetric part. This definition is motivated by the fact that the convergence rate of the generalized minimal residual method for the corresponding linear systems is determined by the ratio. An upper bound for the ratio is established by developing an upper bound for the maximal singular value and a lower bound for the minimal eigenvalue of the symmetric part. It is shown that the bound depends on three factors, the number of the elements in the mesh, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity measured in the metric specified by the inverse diffusion matrix. It is also shown that the diagonal scaling can effectively eliminates the effects from the mesh nonuniformity measured in the Euclidean metric. Numerical results for a selection of examples in one, two, and three dimensions are presented.

math.NA↗

arXiv2018

An adaptive moving mesh finite element solution of the Regularized Long Wave equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a $C^0$ finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.

math.NA↗