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SUMMARYMesh convergence order is a key for certiÿcation of the accuracy of numerical solutions. A few available results and tools in mesh adaption are synthetized in order to specify a mesh converging method for adaptive calculation of compressible ows including shocks or viscous layers. The main design property for this method is early second-order convergence, where early means that the second-order convergence is obtained with coarse meshes.

SUMMARYThe wave equation model, originally developed to solve the advection-diffusion equation, is extended to the multidimensional transport equation in which the advection velocities vary in space and time. The size of the advection term with respect to the diffusion term is arbitrary. An operator-splitting method is adopted to solve the transport equation. The advection and diffusion equations are solved separately at each time step. During the advection phase the advection equation is solved using the wave equation model. Consistency of the first-order advection equation and the second-order wave equation is established. A finite element method with mass lumping is employed to calculate the three-dimensional advection of both a Gaussian cylinder and sphere in both translational and rotational flow fields. The numerical solutions are accurate in comparison with the exact solutions. The numerical results indicate that (i) the wave equation model introduces minimal numerical oscillation, (ii) mass lumping reduces the computational costs and does not significantly degrade the numerical solutions and (iii) the solution accuracy is relatively independent of the Courant number provided that a stability constraint is satisfied.

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