Collection of PhD theses of (former) members of the research group

PhD thesis

vt05 Vera Thümmler.
Numerical analysis of the method of freezing traveling waves


This thesis deals with special solutions of parabolic partial differential equations (PDE), namely traveling waves of the form u(x,t)=w(x-c t). Here w denotes the profile of the wave and c its velocity. The pair (w,c) is an equilibrium of a partial differential algebraic equation (PDAE) which is constructed by inserting the ansatz u(x,t)=v(x-g(t), t) into PDE and adding an additional phase condition. By discretization with finite differences on a finite grid, one obtains a differential algebraic equation (DAE). In the thesis the effect of the transformation PDE -> PDAE (the 'freezing of the wave') and of the discretization PDAE -> DAE to existence and stability of traveling wave solutions, or more general, of relative equilibria, is analyzed. One of the main results is to prove the existence of an equilibrium for DAE which approximates (w,c) and inherits the stability properties of the traveling wave.

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