Second-order short characteristic method for solving the transport equation on a tetrahedron mesh

Abstract

In this paper the second order approximation method on unstructured tetrahedral mesh for solving the transport equation with the help of short characteristics is constructed. The second-order interpolating polynomial is constructed from the values at the vertices of an illuminated face with the use of the values of the integrals of the required function along the edges of the same face. The value at the unilluminated vertex is obtained by integrating along the backward characteristic interval inside the tetrahedron from the interpolated value on the illuminated face. The accuracy of the method depends on the interpolation accuracy and on the source integration along the interval of the characteristic. In the case of piecewise constant approximation of the source part, the method is of the second order, assuming the solution to be sufficiently smooth. On test problems it is shown that the convergence rate of the method is slightly smaller than two in the case of smooth solutions, while this rate is smaller than one for nondifferentiable solution.