an:01417964
Zbl 0953.65065
Marion, M.; Mollard, A.
A multilevel characteristics method for periodic convection-dominated diffusion problems
EN
Numer. Methods Partial Differ. Equations 16, No. 1, 107-132 (2000).
00061314
2000
j
65M25 65M55 65M12 35K15 65M15
multilevel method; spectral method; numerical examples; method of characteristics; periodic solution; convection-diffusion problem; stability; error estimates
The authors consider the linear unsteady convection-diffusion problem that involves a function \(u(x,t)\) from \(]0,2\pi[^d\times \mathbb{R}^+\) into \(\mathbb{R}\):
\[
{\partial u\over\partial t}+ a.\nabla u-\nu\Delta u= f,
\]
where \(\nu\) is the viscosity, \(f\) is the forcing term and \(a\) is some divergence-free vector field. The initial conditions are that \(u(x,0)= u_0(x)\) for all \(x\in ]0,2\pi[^d\) and \(u\) is \(2\pi\)-periodic in all space variables.
The spatial discretization uses the space \(S_M\) of all trigonometric polynomials of degree \(\leq M\) in each variable. A two-level decomposition is introduced by considering another parameter \(m\), \(0< m< M\) and writing \(S_M= S_M= S_m+ (I- P_m)S_M\) where \(P_m\) is the \(L^2\)-projection onto \(S_m\).
The numerical integration then relies on two different interpolation opeators and the approximate solution is obtained as the sum of a term in \(S_m\) and a term in \((I- P_m)S_M\). The two components are advanced in time using different time steps.
The authors investigate the stability fo this scheme and derive error estimates. These indicate that the high-frequency term can be integrated with a larger time-step.
The paper contains a few numerical tests for 1D-problems. It is found that the two-level method allows a significant gain in computing time with respect to the classical method. The accuracy is always better than the one of the classical method based on \(S_m\) and close to the one based on \(S_M\).
Willy Govaerts (Gent)