Variational multi-scale spectral solution of convection-dominated parabolic problems
For researchers in computational fluid dynamics, this provides a method for convection-dominated parabolic problems, but the contribution is incremental as it only extends existing techniques to time-dependent cases with 1D tests.
The paper extends a variational multiscale method with spectral sub-scale approximation to parabolic problems, demonstrating reliability through numerical tests on 1D advection-diffusion-reaction and advection-diffusion equations.
In this work, we consider an extension to parabolic problems of the variational multiscale method with spectral approximation of the sub-scales. We first discretize in time using a finite difference scheme and second, apply the generalization of the spectral variational multi-scale method. To obtain error estimations in convection-dominated flows, we find a helpful link between the stabilized term expressed in terms of Green's functions and in terms of spectral functions. Finally, we present some numerical tests to show the reliability of the method. We consider the stationary one-dimensional advection-diffusion-reaction equation and the evolutive one-dimensional advection-diffusion equation.