High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems

Authors:
Fausto Cavalli;Giovanni Naldi;Gabriella Puppo;Matteo Semplice
Affiliations:
-;-;-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2007

Citing 0
Cited 6

Preconditioned Implicit Solvers for Nonlinear PDEs in Monument Conservation

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High Order Finite Difference WENO Schemes for Nonlinear Degenerate Parabolic Equations

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Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

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Analysis of Multigrid Preconditioning for Implicit PDE Solvers for Degenerate Parabolic Equations

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Numerical Modeling of Degenerate Equations in Porous Media Flow

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Linearly Implicit Approximations of Diffusive Relaxation Systems

Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications

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Abstract

Several relaxation approximations to partial differential equations have been recently proposed. Examples include conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, and gas dynamics problems. The present paper focuses on diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations. These schemes are based on a suitable semilinear hyperbolic system with relaxation terms. High-order methods are obtained by coupling ENO and weighted essentially nonoscillatory (WENO) schemes for space discretization with implicit-explicit (IMEX) schemes for time integration. Error estimates and a convergence analysis are developed for semidiscrete schemes with a numerical analysis for fully discrete relaxed schemes. Various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes, also in the degenerate case. These schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in two- and three-dimensional cases.