Theoretical and numerical analysis on a thermo-elastic system with discontinuities

A second-order accurate numerical scheme is proposed for a thermo-elastic system which models a bar made of two distinct materials. The physical parameters involved may be discontinuous across the joint of the two materials, where there might be also singular heat and/or force sources. The solution components, the temperature and the displacement, may change rapidly across the joint. By transforming the system into a different one, time-marching schemes can be used for the new system which is well posed. The immersed interface method is employed to deal with the discontinuities of the coefficients and the singular sources. The proposed numerical method can fit both explicit and implicit formulation. For the implicit version, a stable and fast prediction-correction scheme is also developed. Convergence analysis shows that our method is second-order accurate at all grid points in spite of the discontinuities across the interface. Numerical experiments are performed to support the theoretical analysis in this paper.