The Numerical Solution of Helmholtz Equation Using Modified Wavelet Multigrid Method

International Journal of Modern Mathematical Sciences, 2020, 18(1): 76-91 International Journal of Modern Mathematical Sciences ISSN:2166-286X Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx Florida, USA Article The Numerical Solution of Helmholtz Equation Using Modified Wavelet Multigrid Method S. C. Shiralashetti1, A. B. Deshi2,*, M. H. Kantli3 1Department of Mathematics, Karnatak University, Dharwad, India-580003 2Department of Mathematics, KLECET, Chikodi, India-591201 3Department of Mathematics, BVV’s BGMIT, Mudhol, India-587313 *Author to whom correspondence should be addressed; E-Mail: [email protected] Article history: Received 11 June 2020, Revised 23 July 2020, Accepted 1 September 2020, Published 14 September 2020. Abstract: This paper presents a modified wavelet multigrid technique for solving elliptic type partial differential equations namely Helmholtz equation. The solution is first obtained on the coarser grid points, and then it is refined by obtaining higher accuracy by increasing the level of resolution. The implementation of the classical numerical methods has been found to involve some difficulty to observe fast convergence in low computational time. To overcome this, we have proposed modified wavelet multigrid method using wavelet intergrid operators similar to classical intergrid operators. Some of the numerical test problems are presented to demonstrate the applicability and attractiveness of the present implementation. Keywords: Wavelet multigrid; Helmholtz equation; Intergrid operators; Numerical methods. Mathematics Subject Classification: 65T60, 97N40, 35J25 1. Introduction The mathematical modelling of engineering problems usually leads to sets of partial differential equations and their boundary conditions. There are several applications of elliptic partial differential equations (EPDEs) in science and engineering. Many physical processes can be modeled using EPDEs. Analytical solution of EPDEs, however, either does not exist or is hard to find. It is precisely due to this Copyright © 2020 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2020, 18(1): 76-91 77 fact that several efficient and accurate methods have been developed for finding numerical solution of EPDEs. Recent contribution in this regard includes several methods such as finite-difference method (FDM), finite element method (FEM) and other methods. In this paper, an alternative method is proposed named as wavelet multigrid (WMG) methods similar to multigrid method for the numerical solution of EPDEs especially the Helmholtz equation [1, 2], 22uu ku(,) x y f (,),0 x y x , y 1, (1.1) xy22 subject to non-homogeneous boundary conditions (BCs). Where k is a constant and f(,) x y is a given non-homogeneous function. Since from the several decades, finite difference methods have been commonly used for the approximate solution of boundary value problems (BVPs) for ordinary and partial differential equations [3]. To seek solutions to differential equations, for most cases, it is necessary to employ discretization methods to reduce the sets of differential equations to systems of algebraic equations. Systems of algebraic equations are related with many problems, as well as with applications of mathematics. Direct methods are used to solve a linear system of N equations with N unknowns. Direct methods are theoretically producing the exact solution to the system in a finite number of steps. In practice, of course, the solution obtained will be polluted by the round-off error. To minimize such round-off error iterative methods are infrequently used for solving linear systems. Since the time required for sufficient accuracy exceeds that required for direct methods. For large systems, these methods are efficient in terms of both computer storage and computation cost. The multigrid method is largely applicable in increasing the efficiency of iterative methods used to solve large system of algebraic equations [4]. The multigrid (MG) method is a well-founded numerical method for solving sparse linear system of equations approximating the differential equations. In the historical three decades the development of effective iterative solvers for systems of algebraic equations has been a significant research topic in numerical analysis and computational science and engineering. For a detailed treatment of multigrid methods we refer Hackbusch [5]. An introduction of multigrid methods is found in Wesseling, Briggs and Trottenberg et al. [6-8]. However, when met by certain problems, the standard multigrid procedure converges slowly with larger computational time. Whereas wavelet multigrid methods solves the system of equations in faster convergence with lesser computational cost [9]. Wavelets have numerous applications in approximation theory and have been extensively used in the context of numerical approximation and also in many areas, such as image processing and time series analysis in the relevant literature during the last two decades. In recent years, wavelet analysis is fast extensive kindness in the numerical solution of elliptic problems. Recently, many authors De Leon [9] and Bujurke et al. [10-12]) have developed wavelet multigrid methods. These methods use a choice Copyright © 2020 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2020, 18(1): 76-91 78 of the filter operators obtained from wavelets to define the prolongation and restriction operators. Avudainayagam and Vani [13] used wavelet-based interpolation and restriction operators for their multigrid approaches, and Vasilyev and Kevlahan [14] used a wavelet-collocation-based multigrid method. Shiralashetti et al. [15] had proposed the modified wavelet multigrid method (MWMG) for the solution of boundary value problems. This paper outspreads the same approach for the numerical solution of Helmholtz equation. The WMG methods formulated in this paper have the following characteristics. They provide approximations which are continuous and continuously differentiable throughout the domain of the problems, and have piecewise continuous second derivatives. The methods possess super convergence properties. The methods incorporate BCs in a systematic fashion. The organization of the rest of the paper is as follows. In section 2, preliminaries of Daubechies wavelets are given. Section 3 describes the method of solution. Numerical examples are presented in section 4. Finally, the conclusions of the proposed work are discussed in section 5. 2. Preliminaries of Daubechies Wavelets A major problem in the growth of wavelets during the 1980’s was the search for a multiresolution analysis where the scaling function was compactly supported and continuous. As already we know that, the Haar multi-resolution analysis is generated by a compactly supported scaling function but it is not continuous. The B-splines are continuous and compactly supported but fail to form an orthonormal basis. A family of multiresolution analyses generated by scaling functions, which are both compactly supported and continuous. These multiresolution analyses were first constructed by Daubechies described in [16] that created great eagerness among mathematicians and scientists performance research in the area of wavelets. 2.1. Multi-resolution Analysis 2 A multi-resolution analysis is basically a set of nested subspaces Vj , j Z of LR(), that obey the following properties: i) ......VVV1 0 1 2 ii) clos2 V L() R L j iii) Vj 0 iv) VVWj1 j j Copyright © 2020 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2020, 18(1): 76-91 79 v) f( x ) Vjj f (2 x ) V1 , j Z Let V the so-called “scaling function” that generates the multi-resolution analysis V of LR2 () 0 j jZ . Then ():k k Z (2.1) is a basis of V0 , and by setting jj/2 jk, (x ) 2 (2 x k ) (2.2) it follows that, for each jZ , the family jk, : kZ (2.3) is also a basis of V j . Then, since V0 is in V1 and since 1, k : kZ is a basis of , there exists a unique sequence ak that describes the following “two-scale relation”: (x ) ak (2 x k ) (2.4) k of the scaling function . Wavelets are functions generated from one single function called the mother wavelet by the simple operations of dilation and translation. A mother wavelet gives rise to a decomposition of the 2 Hilbert space LR(), into a direct sum of closed subspacesWj , j Z [17]. jj/2 Let jk, (x ) 2 (2 x k ) and W clos2 : k Z (2.5) jLR() j, k Then every f L2 () R has a unique decomposition, f()...... x s1 s 0 s 1 (2.6) where sWjj for all jZ , it is, 2 LRWWWW()...... j 1 0 1 (2.7) jZ 2 2 Using this decomposition of LR(), a nested sequence of closed subspaces Vj , j Z of LR() can be obtained, defined by VWWj... j21 j (2.8) 2.2. Daubechies Wavelets Copyright © 2020 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2020, 18(1): 76-91 80 Different choices for may yield different multi-resolution analyses and the most useful scaling functions are those that have compact support. As an example of multiresolution analysis, a family of orthogonal Daubechies wavelets with compact support has been constructed by Daubechies [17]. A wavelet basis is orthonormal if any two translated or dilated wavelets satisfy the condition ()()x x dx (2.9) n,,,, k m l n m k l where is the Kronecker Delta function. Each wavelet family is governed by a set of N (an even integer) coefficients, ak : k 0, 1, . , N 1 through the two-scale relation, N 1 N(x ) a k N (2 x k ) (2.10) k0 Based on the scaling function N ()x , the mother wavelet can be written as, 1 N(x ) b k N (2 x k ) (2.11) kN2 Since the wavelets are orthonormal to the scaling basis the coefficients of the scaling function and the mother wavelet for the two-scale equation are related by: k bak( 1) N1 k (2.12) Daubechies [13] found and exploited the link between vanishing moments of the wavelet and regularity of scaling and wavelet functions, and .

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