High-order relaxation schemes for nonlinear degenerate diffusion problems. (English) Zbl 1171.65065

Cavalli, Fausto; Naldi, Giovanni; Puppo, Gabriella; Semplice, Matteo

SIAM J. Numer. Anal. 45, No. 5, 2098-2119 (2007).

This paper deals with diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations. The main idea is to approximate the original partial differential equation (PDE) with a suitable semilinear hyperbolic system with stiff relaxation terms. As the relaxation parameter is convergent to zero, the solution of the hyperbolic system converges to the solution of the original PDE. An analysis of relaxation schemes, from both the theoretical and computational point of view, is made for the nonlinear degenerate diffusion problem
\[ {\partial u \over \partial t} = D\Delta(p(u)), ~x \in {\mathbb R}^d, ~t>0, \] with initial data in \(L^1(\mathbb R^d)\). The function \(p:{\mathbb R} \rightarrow {\mathbb R}\) is nondecreasing and Lipschitz continuous. The diffusion problem is degenerate if \(p(0)=0\). Error estimates and a convergence analysis are developed for semidiscrete schemes with a numerical analysis for fully discrete relaxed schemes. 1D and 2D numerical results show the high accuracy of the proposed numerical schemes, also in the degenerate case. The authors assert that these numerical schemes can be easily implemented on parallel computers. Numerical results are given for linear diffusion and for porous media equation.

Reviewer: Viorel Arnăutu (Iaşi)

MSC:

65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K65	Degenerate parabolic equations
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
65Y05	Parallel numerical computation

Keywords:

relaxation schemes; high-order accuracy; porous media equation; semidiscretization; parallel computation; nonlinear parabolic equations; nonlinear degenerate diffusion problem; error estimates; convergence; numerical results