Comparing numerical methods for Helmholtz equation model problem

2004 ◽  
Vol 150 (3) ◽  
pp. 763-773 ◽  
Author(s):  
Salah M. El-Sayed ◽  
Doǧan Kaya
Keyword(s):  
Numerical Methods ◽  
Helmholtz Equation ◽  
Model Problem ◽  
Equation Model

Two numerical methods for an inverse problem for the 2-D Helmholtz equation

2003 ◽  
Vol 184 (1) ◽  
pp. 122-148 ◽  
Author(s):  
Yuriy A. Gryazin ◽  
Michael V. Klibanov ◽  
Thomas R. Lucas
Keyword(s):  
Inverse Problem ◽  
Numerical Methods ◽  
Helmholtz Equation

A Contribution to Doubly Asymptotic Approximations: An Operator Top-Down Derivation

1991 ◽  
Vol 113 (3) ◽  
pp. 409-415 ◽  
Author(s):  
B. Nicolas-Vullierme
Keyword(s):  
Steady State ◽  
Helmholtz Equation ◽  
Model Problem ◽  
Wave Number ◽  
Asymptotic Approximations ◽  
Exterior Problem ◽  
Top Down ◽  
High K ◽  
Low K ◽  
Order Continuous

As a contribution to the theory of Doubly Asymptotic Approximations (DAAs), a formal operator top-down derivation specialized to steady-state motions is proposed for the Neumann exterior problem associated to the Helmholtz equation. It generalizes previously published ones which relied either on a modal approach [1] or on a scalar approach for a model problem [2]. The proposed derivation is based on an integral representation of the solution of the Helmholtz equation in an unbounded domain: first, two asymptotic expansions of the representation with respect to the nondimensional wave number k are obtained in the low-k and the high-k ranges; then these expansions are matched. This procedure allows to point out that some geometrical physical and mathematical assumptions underlie the validity of high order continuous forms of the DAAs. Special attention is then devoted to their discretized counterparts which are compared to previously published ones. In both cases it suggests further investigation of some interesting and open geometry and numerical analysis problems.

scholarly journals The Necessary Conditions Imposed on the Coefficients of Multistep Hybrid Methods Adapted for Solving ODEs of the Second Order

2021 ◽  
Vol 20 ◽  
pp. 344-352
Author(s):  
Vusala Nuriyeva
Keyword(s):  
Numerical Methods ◽  
Computer Technology ◽  
Hybrid Method ◽  
Initial Value Problem ◽  
Necessary Conditions ◽  
Hybrid Methods ◽  
Model Problem ◽  
Second Order ◽  
Initial Value ◽  
Theoretical Results

There are some classes of methods to solve the initial-value problem for the ODEs of the second order. Recently among of them are developed the numerical methods, which are using in the application of computer technology. By taking into account the wide application of the numerical methods, here has investigated the numerical solution of the above-mentioned problem. For this aim here has constructed the multistep hybrid method with the special structure, which has been applied to solve the initial-value problem of the ODEs of the second order. Given some recommendation to choosing of the suitable methods for solving above named problem and also, have found some bounders imposed on the coefficients of the convergence methods. Constructed specific methods solve the initial-value problem for ODEs of the second order. The received theoretical results have been illustrated by using some concrete methods, which have applied to solve model problem for ODEs of the second order

scholarly journals The implementation of the boundary element method to the Helmholtz equation of acoustics

2021 ◽  
pp. 13-19
Author(s):  
Sergey Sivak ◽  
Mihail Royak ◽  
Ilya Stupakov ◽  
Aleksandr Aleksashin ◽  
Ekaterina Voznjuk
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Element Method ◽  
Numerical Methods ◽  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Boundary Element ◽  
Three Dimensional ◽  
The Finite Element Method ◽  
Element Method

Introduction: To solve the Helmholtz equation is important for the branches of engineering that require the simulation of wave phenomenon. Numerical methods allow effectiveness’ enhancing of the related computations. Methods: To find a numerical solution of the Helmholtz equation one may apply the boundary element method. Only the surface mesh constructed for the boundary of the three-dimensional domain of interest must be supplied to make the computations possible. This method’s trait makes it possible toconduct numerical experiments in the regions which are external in relation to some Euclidian three-dimensional subdomain bounded in the three-dimensional space. The later also provides the opportunity of not using additional geometric techniques to consider the infinitely distant boundary. However, it’s only possible to use the boundary element methods either for the homogeneous domains or for the domains composed out of adjacent homogeneous subdomains. Results: The implementation of the boundary elementmethod was committed in the program complex named Quasar. The discrepancy between the analytic solution approximation and the numerical results computed through the boundary element method for internal and external boundary value problems was analyzed. The results computed via the finite element method for the model boundary value problems are also provided for the purpose of the comparative analysis done between these two approaches. Practical relevance: The method gives an opportunityto solve the Helmholtz equation in an unbounded region which is a significant advantage over the numerical methods requiring the volume discretization of computational domains in general and over the finite element method in particular. Discussion: It is planned to make a coupling of the two methods for the purpose of providing the opportunity to conduct the computations in the complex regions with unbounded homogeneous subdomain and subdomains with substantial inhomogeneity inside.

scholarly journals ROBUST NUMERICAL METHODS FOR BOUNDARY‐LAYER EQUATIONS FOR A MODEL PROBLEM OF FLOW OVER A SYMMETRIC CURVED SURFACE

2006 ◽  
Vol 11 (4) ◽  
pp. 365-378
Author(s):  
A. R. Ansari ◽  
B. Hossain ◽  
B. Koren ◽  
G. I. Shishkin
Keyword(s):  
Boundary Layer ◽  
Numerical Methods ◽  
Numerical Method ◽  
Original Problem ◽  
Model Problem ◽  
Curved Surface ◽  
Finite Domain ◽  
Third Order ◽  
Boundary Layer Equations ◽  
Self Similar

We investigate the model problem of flow of a viscous incompressible fluid past a symmetric curved surface when the flow is parallel to its axis. This problem is known to exhibit boundary layers. Also the problem does not have solutions in closed form, it is modelled by boundary‐layer equations. Using a self‐similar approach based on a Blasius series expansion (up to three terms), the boundary‐layer equations can be reduced to a Blasius‐type problem consisting of a system of eight third‐order ordinary differential equations on a semi‐infinite interval. Numerical methods need to be employed to attain the solutions of these equations and their derivatives, which are required for the computation of the velocity components, on a finite domain with accuracy independent of the viscosity v, which can take arbitrary values from the interval (0,1]. To construct a robust numerical method we reduce the original problem on a semi‐infinite axis to a problem on the finite interval [0, K], where K = K(N) = ln N. Employing numerical experiments we justify that the constructed numerical method is parameter robust.

scholarly journals SIXTH-ORDER ACCURATE FINITE DIFFERENCE SCHEMES FOR THE HELMHOLTZ EQUATION

2006 ◽  
Vol 14 (03) ◽  
pp. 339-351 ◽  
Author(s):  
I. SINGER ◽  
E. TURKEL
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Truncation Error ◽  
Model Problem ◽  
Difference Schemes ◽  
Sixth Order ◽  
Finite Difference Schemes ◽  
Local Truncation Error ◽  
Two Dimensional ◽  
Truncation Order

We develop and analyze finite difference schemes for the two-dimensional Helmholtz equation. The schemes which are based on nine-point approximation have a sixth-order accurate local truncation order. The schemes are compared with the standard five-point pointwise representation, which has second-order accurate local truncation error and a nine-point fourth-order local truncation error scheme based on a Padé approximation. Numerical results are presented for a model problem.

A Comparison Study of Numerical Methods for Compressible Two-Phase Flows

2017 ◽  
Vol 9 (5) ◽  
pp. 1111-1132 ◽  
Author(s):  
Jianyu Lin ◽  
Hang Ding ◽  
Xiyun Lu ◽  
Peng Wang
Keyword(s):  
Numerical Methods ◽  
Numerical Experiments ◽  
Mass Conservation ◽  
Detailed Comparison ◽  
Equation Model ◽  
Bubble Collapse ◽  
Comparison Study ◽  
Two Phase Flows ◽  
Two Phase ◽  
Interface Position

AbstractIn this article a comparison study of the numerical methods for compressible two-phase flows is presented. Although many numerical methods have been developed in recent years to deal with the jump conditions at the fluid-fluid interfaces in compressible multiphase flows, there is a lack of a detailed comparison of these methods. With this regard, the transport five equation model, the modified ghost fluid method and the cut-cell method are investigated here as the typical methods in this field. A variety of numerical experiments are conducted to examine their performance in simulating inviscid compressible two-phase flows. Numerical experiments include Richtmyer-Meshkov instability, interaction between a shock and a rectangle SF6 bubble, Rayleigh collapse of a cylindrical gas bubble in water and shock-induced bubble collapse, involving fluids with small or large density difference. Based on the numerical results, the performance of the method is assessed by the convergence order of the method with respect to interface position, mass conservation, interface resolution and computational efficiency.

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