Higher-order boundary element methods (BEM) are presented for time-dependent convective diffusion in two dimensions. The time-dependent convective diffusion free-space fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. Boundary element method solutions up to the Peclet number 106 are obtained for an example problem of unsteady convection-diffusion that possesses an exact solution. We investigate the convergence rate and accuracy of the higher-order boundary element formulations. An extremely high accuracy of the BEM solutions for highly convective flows is demonstrated. Moreover, it is shown that the use of time-dependent convective kernels provides an automatic upwinding for the entire range of Peclet numbers and also leads to very efficient algorithms as the Peclet number increases.

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