A high-order numerical approach with Cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials

Recently we have developed a new numerical approach for PDEs with constant coefficients on irregular domains and Cartesian meshes. In this paper we extend it to a much more general case of PDEs with variable coefficients that have a lot of applications; e.g., the modeling of functionally graded materials, the inhomogeneous materials obtained by 3-D printing and many others. Here, we consider the 2-D wave and heat equations for isotropic and anisotropic inhomogeneous materials. The idea of the extension to the case of PDEs with variable coefficients is based on the representation of the stencil coefficients as functions of the mesh size. This leads to the increase in the size of the local system of algebraic equations solved for each grid point of the new approach; however, this does not change the size of the global system of semidiscrete equations and practically does not increase the computational costs of the proposed technique. Similar to our previous technique, the new 2-D approach with compact 9-point stencils uses trivial Cartesian meshes for complex irregular domains and provides the fourth order of accuracy for the wave and heat equations with variable coefficients. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. At similar 9-point stencils, the accuracy of the new approach is much higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the third order) finite elements with much wider stencils. The wave and heat equations are uniformly treated with the new approach.