Kinetic flux-vector splitting schemes for the hyperbolic heat conduction

A kinetic solver is developed for the initial and boundary value problems of the symmetric hyperbolic moment system. This nonlinear system of equations is related to the heat conduction in solids at low temperatures. The system consists of a conservation equation for the energy density e and a balance equation for the heat flux Qi, where e and Qi are the four basic fields of the theory. We use kinetic flux vector splitting (KFVS) scheme to solve these equations in one and two space dimensions. The flux vectors of the equations are splitted on the basis of the local equilibrium distribution of phonons. The resulting computational procedure is efficient and straightforward to implement. The second-order accuracy of the scheme is achieved by using MUSCL-type reconstruction and min-mod nonlinear limiters. The solutions exhibit second-order accuracy and satisfactory resolution of gradients with no spurious oscillations. The scheme is extended to the two-dimensional case in a usual dimensionally split manner. In order to prescribe the boundary data, we need the knowledge of the e and Qi. However, in experiments only one of the quantities can be controlled at the boundary. This problem is removed by using a continuity condition. It turned out that after some short time energy and heat flux are related to each other according to Rankine–Hugoniot jump relations. To illustrate the performance of the KFVS scheme, we perform several one- and two-dimensional test computations. For the comparison of our results, we use high order central schemes. The present study demonstrates that this kinetic method is effective in handling such problems.