Abstract
In this paper, an HOC scheme with multigrid algorithm is developed for solving the Cauchy problem associated with two dimensional Helmholtz type equations. The suggested scheme has up to fourth order accuracy. Lastly, some numerical experiments are given to show the accuracy and performance of the proposed scheme.
Highlights
Helmholtz equation has many real-world applications related to wave propagation and vibrating phenomena [1], the radiation and scattering of wave [2, 3]
The main focus in this paper is to develop multigrid method based on Higher order compact schemes (HOC) scheme on non-uniform grids for solving of 3D Helmholtz equation
We have studied the Cauchy problem for Helmholtz type equation in two and three-dimensional cases
Summary
Helmholtz equation has many real-world applications related to wave propagation and vibrating phenomena [1], the radiation and scattering of wave [2, 3]. The boundary conditions are often incomplete in many engineering problems and the solution is prescribed at some interior points in the domain. These are called ill-posed problems such that the stability and uniqueness of their solution are not guaranteed [7]. A classic example of an inverse problem for Helmholtz-type equation is the Cauchy problem. In this problem, the boundary conditions for both the solution and its normal derivative are prescribed only on a part of the boundary of the solution domain while having no information about the remaining part of the boundary. There has recently devoted to the search for better and more efficient methods for determining an approximate or numerical solution
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