Numerical solution of differential and integral equations
The paper is devoted to constructing local absorbing boundary conditions (ABCs) for numerical solution to the n-dimensional wave equation in an arbitrary open domain with a piecewise smooth artificial boundary. Our method is essentially based on the use of the techniques of operator factorisation, dimensional splitting and spline approximations. Starting from the operator factorisation and employing the operator splitting by coordinates, we reduce the construction of ABCs for the original n-D equation to seeking for an ABC for the one-dimensional wave problem. Further, applying the Laplace transform in time and approximating the initial data by a family of splines, we obtain an infinite hierarchy of functions that approximate the far-field solution with increasing accuracy. These functions are then used as ABCs. Due to the property of compactness of supports of the splines, the boundary conditions appear to be local both in time and space, which is extremely important for the numerical calculations. Moreover, these ABCs are absolutely uncritical to the shape of the artificial boundary, and therefore, usable for solving practical wave problems in domains of drastically complex geometries. The resulting boundary value problems are well-posed in the sense of existence, uniqueness and stability of solution. Results of the numerical experiments confirm the functionality and efficiency of the approach.
Note. Abstracts are published in author's edition
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