The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations

DOI10.1016/j.jcp.2013.01.052zbMath1297.65097arXiv1206.6312OpenAlexW2023111538MaRDI QIDQ401511

Erik S. Van Vleck, Changna Lu, Weizhang Huang

Publication date: 27 August 2014

Published in: Journal of Computational Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1206.6312

zbMATH Keywords

cutoffconvergenceparabolic partial differential equationsnonnegative solutionfinite elementerror analysisprojectionpositive solutionlinearcollocationfinite volumefinite differencespectral methodanisotropic diffusionnumerical resultlubrication-type equation

Mathematics Subject Classification ID

Nonlinear parabolic equations (35K55) Initial-boundary value problems for second-order parabolic equations (35K20) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12)

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Uses Software

• MOVCOL4

Cites Work

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• A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem
• Higher order nonlinear degenerate parabolic equations
• The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem
• On maximum-principle-satisfying high order schemes for scalar conservation laws
• A linear scheme satisfying a maximum principle for anisotropic diffusion operators on distorted grids
• A mixed implicit-explicit finite difference scheme for heat transport in magnetised plasmas
• A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems
• On maximum principles for monotone matrices
• Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy
• Spatial coupling of plant and herbivore dynamics: The contribution of herbivore dispersal to transient and persistent ``waves of damage
• Discrete maximum principle for linear parabolic problems solved on hybrid meshes
• Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes
• Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal hardening
• An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems
• Preserving monotonicity in anisotropic diffusion
• Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes
• On discrete maximum principles for nonlinear elliptic problems
• Modelling of heat transport in magnetised plasmas using non-aligned coordinates
• Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients
• Maximum principle in linear finite element approximations of anisotropic diffusion-convection-reaction problems
• Discrete maximum principle for finite-difference operators
• Maximum principle and uniform convergence for the finite element method
• Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes. (Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés).
• Maximum Principles for $P1$-Conforming Finite Element Approximations of Quasi-linear Second Order Elliptic Equations
• Discrete Maximum Principle and a Delaunay-Type Mesh Condition for Linear Finite Element Approximations of Two-Dimensional Anisotropic Diffusion Problems
• Discrete maximum principles for FE solutions of nonstationary diffusion-reaction problems with mixed boundary conditions
• An Algebraic Discrete Maximum Principle in Hilbert Space with Applications to Nonlinear Cooperative Elliptic Systems
• Diagonally Implicit Runge-Kutta Formulae with Error Estimates
• Three-Dimensional Delaunay Triangulations for Finite Element Approximations to a Second-Order Diffusion Operator
• Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s
• On the motion of a small viscous droplet that wets a surface
• A monotone finite element scheme for convection-diffusion equations
• Convergence of an Iterative Method for Total Variation Denoising
• Numerical Solution of Partial Differential Equations
• Failure of the discrete maximum principle for an elliptic finite element problem
• On the Cahn–Hilliard Equation with Degenerate Mobility
• Symmetric Singularity Formation in Lubrication-Type Equations for Interface Motion
• Positivity-Preserving Numerical Schemes for Lubrication-Type Equations
• Discrete Maximum Principle and Adequate Discretizations of Linear Parabolic Problems
• On the existence of maximum principles in parabolic finite element equations
• MOVCOL4: A Moving Mesh Code for Fourth‐Order Time‐Dependent Partial Differential Equations
• Une méthode d'approximation de la solution des équations de Navier-Stokes
• Numerical Solution of the Navier-Stokes Equations