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A multilevel characteristics method for periodic convection-dominated diffusion problems. (English) Zbl 0953.65065
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 $$S_M$$.

##### MSC:
 65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs 65M55 Multigrid methods; domain decomposition 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 35K15 Initial value problems for second-order parabolic equations 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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