Computational Mathematics and Scientific Computing Seminar

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New Numerical Methods for Solving the Advection Equation and Wave Propagation Problems

Speaker: Haruhiko Kohno, Lehigh University

Location: Warren Weaver Hall 1302

Date: Feb. 24, 2012, 10 a.m.

Synopsis:

In this presentation, I would like to talk about two recently-developed numerical methods, namely the hierarchical-gradient truncation and remapping (H-GTaR) method and finite element wave-packet method. The former method is a new type of Semi-Lagrangian method for solving the advection equation. Our strategy reduces the original PDE to a set of decoupled linear ODEs with constant coefficients. Additionally, we introduce a remapping strategy to periodically guarantee solution accuracy for a deformation problem. The proposed scheme yields nearly an exact solution for a rigid body motion with a smooth function that possesses vanishingly small higher derivatives and calculates the gradient of the advected function in a straightforward way. The proposed method is verified through several 2D classical benchmark problems. The latter method was developed for the purpose of solving for multiscale plasma waves in a tokamak poloidal plane accurately with reasonable computational cost, but the proposed method can also be applicable to any multiscale wave problems. This method is established by combining the advantages of the finite element and spectral methods, so that important properties in the finite element method, such as the sparsity of the global matrix and the ease in satisfying the boundary conditions, are retained. The present scheme is applied to some illustrative 1D multiscale problems, and its accuracy improvement is demonstrated through comparisons with the conventional finite element method.