A Eulerian-Lagrangian numerical scheme for the dispersion-convection equation using conjugate space-time grids

Abstract A new numerical scheme is proposed for the dispersion-convection equation which combines the utility of a fixed grid in Eulerian coordinates with the computational power of the Lagrangian method. Convection is formally decoupled from dispersion in a manner which does not leave room for ambiguity. The resulting convection problem is solved by the method of characteristics on a grid fixed in space. Dispersion is handled by finite elements on a separate grid which may, but need not, coincide wit the former at selected points in spacetime. Information is projected from one grid to another by local interpolation. The conjugate grid method is implemented by linear finite elements in conjunction with piecewise linear interpolation functions and applied to five problems ranging from predominant dispersion to pure convection. The results are entirely free of oscillations. Numerical dispersion exists but can be brought under control either by reducing the spatial increment, or by increasing the time step size, of the grid used to solve the convection problem. Contrary to many other methods, best results are often obtained when the Courant number exceeds 1.

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