Eighth International Undergraduate Summer Research Symposium
نویسندگان
چکیده
This project’s aim seeks to study the acceleration of numerical algorithms used to solve the Helmholtz equation with Dirichlet boundary conditions. The Helmholtz equation arises naturally in physical problems modeled by partial differential equations in space and time as a time-independent wave equation when applying the technique of separation of variables. Using Dirichlet boundary conditions on an infinite domain helps in modelling an outgoing scattered wave after incidence without transmission through the reflecting media. In this work, we are interested in robust numerical methods based on integral equations to solve the Helmholtz equation. It is known that this technique leads to large and dense linear systems in particular in the high-frequency regime, therefore, it is difficult to solve even when using High Performance Computing. We are investigating here some iterative methods based on Domain Decomposition Algorithms to solve the so-called single layer potential. The main idea consists of splitting the boundary of the obstacle into many sub-boundaries and solving the integral equation separately on each of these surfaces. This procedure is performed at each iteration, where some of the data are exchanged between these surfaces in order to recover the original solution. This employs a transfer-matrix method to solve for the density function as a vector in a matrix equation by LU factorization. In using this technique, we hope to see accelerating convergence of the approximations to the density function in the near future. Helmholtz Equation Dirichlet Boundary Conditions ∆u + ku = 0 u(R, θ) = f(R) limr→∞(|u(r, θ)|) < ∞ Single Layer Potential in terms of Green’s function and density ρ(τ) u(x) = ∫ G(t, τ)ρ(τ)dτ = −e ∂Ω
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تاریخ انتشار 2015