Bicompact interpolation-characteristic scheme of the third order of approximation for the linear transport equation

Abstract:

An explicit interpolation-characteristic scheme is constructed for the numerical solution of the homogeneous linear transport equation. In this scheme, the value of the unknown function at a node on the upper time layer is equal to the value of the function at the intersection point of the characteristic passing through this node with the lower time layer. In turn, the latter value is estimated using a local Hermite interpolation polynomial built in a spatial cell between two integer nodes where the intersection point is located. In this paper, in contrast to other works, the values of the unknown function in two neighboring integer nodes are used to construct the Hermite polynomial, along with the values of its primitive, and not the values of its derivative. The constructed scheme is bicompact and is intended primarily for computing transfer problems with nonsmooth solutions. Computations of solutions of test problems with different smoothness of initial data on refining nested grids are performed. These calculations, in particular, showed the third order of convergence of the bicompact scheme in the case of smooth solutions. The dissipative and dispersive properties of the bicompact scheme are investigated.