The second order short-characteristics method for the solution of the transport equation on a tetrahedron grid

In this paper the second order approximation method on unstructured tetrahedral mesh for solving the transport equation by the use of short characteristics is constructed. Second order polynomial interpolation constructed by the values at the tops of the illuminated face and the values of the integrals of the unknown function along the edges of the same face. The value in the nonilluminated top is obtained by integrating along the characteristic inside the tetrahedron from the interpolated value on the illuminated face. Accuracy of the method depends on the interpolation accuracy and the accuracy of the right part integration along the segment of the characteristic. In the case of piecewise constant approximation of the right part it is the second order of convergence on the condition that the solution has sufficient smoothness. On the test problems it is shown that in the case of smooth solutions the method has the order of convergence a little less than second, in the case of non-differentiable solution — lesser than first.