A third-order projection-characteristic method for solving the transport equation on unstructed grids

The paper presents a numerical study of the convergence order of the projection-characteristic СРР (Cubic Polynomial Projection) method for solving a three-dimensional stationary transport equation on unstructured tetrahedral meshes. The method is based on a characteristic approach to solving the transport equation, has a minimal stencil within a single tetrahedron and a high (third) order of approximation. Unlike classical grid-characteristic methods, in this method, the final numerical approach is constructed not on the basis of interpolation operators of some order of approximation, but on the basis of orthogonal projection operators on the functional space used to approximate the solution. The base scheme is a one-dimensional scheme referred to the Hermitian cubic Interpolation СIP (Cubic Interpolation Polynomial) scheme. The use of interpolation operators is often implemented to sufficiently smooth functions. However, even if the exact solution has sufficient smoothness, some types of illumination of tetrahedra lead to the appearance of non-smooth grid solutions. The transition to orthogonal projectors solves two problems: firstly, the problem of the appearance of angular directions that are coplanar with the faces of the cells, and secondly, the problem of the appearance of non-smooth numerical solutions in the faces of the mesh cell. The convergence result is compared with the theoretical estimates obtained for the first time by one of the authors of this work. The third order of convergence of the method is shown, provided that the solution is sufficiently smooth and the absorption coefficient in the cells is near to constant.