Numerical study of a new method to the solution of partial dierential equations on irregular domains using Cartesian meshes



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A new numerical technique for the time dependent wave and heat equations as well as for the time independent Laplace and Helmholtz equations on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point (in the 2-D) and 27-point (in the 3-D) uniform and non-uniform stencil equations (similar to the stencils in linear quadrilateral finite elements) are used for irregular domains. 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 for the new technique. At the same width of the stencil equations of linear quadrilateral fi nite element, the accuracy of the new approach is two orders higher for the Dirichlet boundary conditions and one order higher for the Neumann boundary conditions compared to those for the linear finite elements. Very small distances between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. 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 fi nite elements with much wider stencils. The order of the time derivative in the time dependent equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The Helmholtz and screened Poisson equations can be uniformly treated with the new approach. The new approach can be directly applied to other partial differential equations as well.



Numerical approach, Cartesian mesh, Irregular domains, High-order accuracy, Wave equation, Heat equation, Laplace equation, Helmholtz equation