Variable Wavenumber Research Articles

Approximations of the Helmholtz equation with variable wave number in one dimension

Approximations of the Helmholtz equation with variable wave number in one dimension

A phase velocity preserving fourth-order finite difference scheme for the Helmholtz equation with variable wavenumber

Numerically solving the Helmholtz equation with large wavenumbers can be challenging due to the highly oscillatory nature of the solution. This paper proposes a 3-point finite difference scheme for numerically solving the 1D Helmholtz equation with variable wavenumber. The proposed scheme preserves the phase velocity, making it well-suited for problems with large wavenumbers. Convergence analysis shows that the proposed scheme is guaranteed to have fourth-order accuracy. Numerical results demonstrate that the proposed scheme maintains high accuracy even for problems with large wavenumbers.

A sixth order quasi-compact finite difference method for Helmholtz equations with variable wave numbers

A sixth order quasi-compact finite difference (SQC) scheme for the Helmholtz equation with variable wave numbers in two and three dimensional rectangular domains has been developed and analyzed. The finite difference coefficients for the solution and the weights for the source term have explicit expressions without involving derivatives of the source term. The proof of the sixth order convergence of the new method is also presented. Numerical experiments presented in this paper confirmed the order of convergence of the new method.

BARYCENTRIC RATIONAL INTERPOLATION METHOD OF THE HELMHOLTZ EQUATION WITH IRREGULAR DOMAIN

In the work, a numerical method of the 2D Helmholtz equation with meshless interpolation collocation method is developed, which is defined in arbitrary domain with irregular shape. In our numerical method, based on the Chebyshev points, the partial derivatives and the spatial variables are discretized by the barycentric rational form basis function. After that the differential equations are simplified by employing differential matrix. To verify the the accuracy, effectiveness and stability in our method, some numerical tests based on the three types of different test points are adopted. Moreover, we can also verify that present method can be applied to both variable wave number problems and high wave number problems.

A generalized plane wave discontinuous Galerkin method for three-dimensional anisotropic Helmholtz equations with variable wave numbers

In this paper we are concerned with the numerical method for three-dimensional anisotropic Helmholtz equations with variable wave numbers, where positive definite matrices define anisotropic media. We define novel generalized plane wave basis functions based on rigorous choice of the coordinate transformation. Then we derive the desired error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices, under an assumption on the shape regularity of polyhedral meshes. Numerical results verify the validity of the theoretical results, and indicate that the approximate solutions generated by the proposed method possess high accuracies.

Dirac assisted tree method for 1D heterogeneous Helmholtz equations with arbitrary variable wave numbers

In this paper we introduce a new method called the Dirac Assisted Tree (DAT) method, which can handle 1D heterogeneous Helmholtz equations with arbitrarily large variable wave numbers. DAT breaks an original global problem into many parallel tree-structured small local problems, which are linked together to form a global solution by solving small linking problems. To solve the local problems in DAT, we propose a compact finite difference method (FDM) with arbitrarily high accuracy order and low numerical dispersion for piecewise smooth coefficients and variable wave numbers. This compact FDM is particularly appealing for DAT, because the local problems and their fluxes in DAT can be computed with high accuracy. DAT with such compact FDMs can solve heterogeneous Helmholtz equations with arbitrarily large variable wave numbers accurately by solving small linear systems — 4×4 matrices in the extreme case — with tridiagonal coefficient matrices in a parallel fashion. Several numerical examples are provided to illustrate the effectiveness of DAT using the Mth order compact FDMs with M=6,8 for numerically solving heterogeneous Helmholtz equations with variable wave numbers. We shall also discuss how to solve some special 2D Helmholtz equations using DAT.

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

PurposeThis meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.Design/methodology/approachIn this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.FindingsThe numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.Research limitations/implicationsCompared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.Practical implicationsThe Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.Originality/valueThis meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

A Meshless Collocation Method with Barycentric Lagrange Interpolation for Solving the Helmholtz Equation

In this paper, Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation. First of all, the interpolation basis function is applied to treat the spatial variables and their partial derivatives, and the collocation method for solving the second order differential equations is established. Secondly, the differential equations on a given test node. Finally, based on three kinds of test nodes, numerical experiments show that the present scheme can not only calculate the high wave numbers problems, but also calculate the variable wave numbers problems. In addition, the algorithm has the advantages of high calculation accuracy, good numerical stability and less time consuming.

Propagation Mode Analysis on H2–Air Rotating Detonation Waves in a Hollow Combustor

An experimental study on the propagation mode of –air rotating detonation waves (RDWs) under atmospheric backpressure in a hollow combustion chamber was performed. Experiments were conducted by holding the equivalent ratio at and the mass flow rate of , that is, mass flow rate per unit area of . High-frequency pressure results indicated that increasing mass flow rate shifted the detonation propagation mode from a dual-wave mode with quasi-detonation and detonation mutual transition (comprising three submodes: periodically coupling oscillation, random coupling oscillation, and coupling oscillation with variable wave number) to a single-wave mode with strong instability and finally to a stable single-wave mode. The predetonator was found to play an important role in the stability of detonation propagation. It could bring about injection instability that interacts with the RDWs. Therefore, this interaction, as well as the detonation strength, affects the propagation mode of RDWs. If the predetonator resonated longitudinally, it largely impacts the RDW propagation mode. In the single-wave mode with a lower (e.g., ), the injection instability may cause the propagation frequency to increase, but from the long-term trend, the detonation propagation stability is enhanced with an increase of .

Generalized plane wave discontinuous Galerkin methods for nonhomogeneous Helmholtz equations with variable wave numbers

ABSTRACT In this paper we are concerned with numerical method for nonhomogeneous Helmholtz equations with variable wave numbers. We derive generalized plane wave basis functions for three-dimensional homogeneous Helmholtz equations with variable wave numbers. Then, by combining the local spectral element method, we design a generalized plane wave discontinuous Galerkin method for the discretization of such nonhomogeneous Helmholtz equations (in both dimensions two and three dimensions). We show that the approximation solutions generated by the proposed discretization method yield error estimates with high accuracies. Numerical results indicate that the resulting approximate solutions generated by the new method possess high accuracy and verify the validity of the theoretical results.

A Variable Wave Number Plane Wave Enriched Partition of Unity Isogeometric Analysis for Acoustic Problems in Heterogeneous Media

A Variable Wave Number Plane Wave Enriched Partition of Unity Isogeometric Analysis for Acoustic Problems in Heterogeneous Media

High Order Compact Scheme and the Moving Mesh Method for Helmholtz Equation with Variable Wave Numbers

High order compact scheme has extensively been used for solving the Helmholtz equation on uniform mesh. Although uniform spacing is computationally advantageous for isotropic problems, the non-uniform mesh will be superior to the problems with the anisotropic properties or boundary layers. In this paper, we proposed a compact scheme on nonuniform mesh which has at least third order accuracy for the 2D Helmholtz equation with variable wave number. A novel adaptive mesh algorithm is also presented to generate the orthogonal nonuniform mesh. The 2D Helmholtz equation is solved by combining the HOC scheme on nonuniform mesh with the moving mesh method. The multigrid method is used to solve the discrete algebraic system. Numerical experiments are executed to verify the accuracy and efficiency of the presented method.

Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation

Many important physical phenomena can be described by the Helmholtz equation. We investigate to what extent the convergence of the shifted Laplacian preconditioner for the Helmholtz equation can be accelerated using deflation with multigrid vectors. We therefore present a unified framework for two published algorithms. The first deflates the preconditioned operator and requires no further preconditioning. The second deflates the original operator and combines deflation and preconditioning in a multiplicative fashion. We pursue two scientific contributions. First we show, using a model problem analysis, that both algorithms cluster the eigenvalues. The new and key insight here is that the near-kernel of the coarse grid operator causes a limited set of eigenvalues to shift away from the center of the cluster with a distance proportional to the wave number. This effect is less pronounced in the first algorithmic variant at the expense of a higher computational cost. In the second contribution we quantify for the first time the large amount of reduction in CPU-time that results from the clustering of eigenvalues and the reduction in iteration count. We report to this end on the findings of an implementation in PETSc on two and three-dimensional problems with constant and variable wave number.

Compact high order schemes with gradient-direction derivatives for absorbing boundary conditions

We consider several compact high order absorbing boundary conditions (ABCs) for the Helmholtz equation in three dimensions. A technique called “the gradient method” (GM) for ABCs is also introduced and combined with the high order ABCs. GM is based on the principle of using directional derivatives in the direction of the wavefront propagation. The new ABCs are used together with the recently introduced compact sixth order finite difference scheme for variable wave numbers. Experiments on problems with known analytic solutions produced very accurate results, demonstrating the efficacy of the high order schemes, particularly when combined with GM. The new ABCs are then applied to the SEG/EAGE Salt model, showing the advantages of the new schemes.

The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber

Numerical dispersion, or what is often referred to as the pollution effect, presents a challenge to an efficient finite element discretization of the Helmholtz equation in the medium frequency regime. To alleviate this effect and improve the unsatisfactory pre-asymptotic convergence of the classical Galerkin finite element method based on piecewise polynomial basis functions, several discretization methods based on plane wave bases have been proposed. Among them is the discontinuous enrichment method that has been shown to offer superior performance to the classical Galerkin finite element method for a number of constant wavenumber Helmholtz problems and has also outperformed two representative methods that use plane waves – the partition of unity and the ultra-weak variation formulation methods. In this paper, the discontinuous enrichment method is extended to the variable wavenumber Helmholtz equation. To this effect, the concept of enrichment functions based on free-space solutions of the homogeneous form of the governing differential equation is enlarged to include free-space solutions of approximations of this equation obtained in this case by successive Taylor series expansions of the wavenumber around a reference point. This leads to plane wave enrichment functions based on the piece-wise constant approximation of the wavenumber, and to Airy wave enrichment functions. Several elements based on these enrichment functions are constructed and evaluated on benchmark problems modeling sound-hard scattering by a disk submerged in an acoustic fluid where the speed of sound varies in space. All these elements are shown to outperform by a substantial margin their continuous polynomial counterparts.

Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number

Several studies have presented compact fourth order accurate finite difference approximation for the Helmholtz equation in two or three dimensions. Several of these formulae allow for the wave number k to be variable. Other papers have extended this further to include variable coefficients within the Laplacian which models non-homogeneous materials in electromagnetism.Later papers considered more accurate compact sixth order methods but these were restricted to constant k. In this paper we extend these compact sixth order schemes to variable k in both two and three dimensions. Results on 2D and 3D problems with known analytic solutions verify the sixth order accuracy. We demonstrate that for large wave numbers, the second order scheme cannot produce comparable results with reasonable grid sizes.

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

The method of difference potentials was originally proposed by Ryaben'kii and can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the level of a finite-difference scheme on a regular structured grid. The method of difference potentials assembles the overall solution of the original boundary value problem by repeatedly solving an auxiliary problem. This auxiliary problem allows a considerable degree of flexibility in its formulation and can be chosen so that it is very efficient to solve. Compact finite difference schemes enable high order accuracy on small stencils at virtually no extra cost. The scheme attains consistency only on the solutions of the differential equation rather than on a wider class of sufficiently smooth functions. Unlike standard high order schemes, compact approximations require no additional boundary conditions beyond those needed for the differential equation itself. However, they exploit two stencils--one applies to the left-hand side of the equation and the other applies to the right-hand side of the equation. We shall show how to properly define and compute the difference potentials and boundary projections for compact schemes. The combination of the method of difference potentials and compact schemes yields an inexpensive numerical procedure that offers high order accuracy for non-conforming smooth curvilinear boundaries on regular grids. We demonstrate the capabilities of the resulting method by solving the inhomogeneous Helmholtz equation with a variable wavenumber with high order (4 and 6) accuracy on Cartesian grids for non-conforming boundaries such as circles and ellipses.

DEVELOPMENT OF A THREE-POINT SIXTH-ORDER HELMHOLTZ SCHEME

A compact finite-difference scheme is presented in this paper for efficiently solving the variable Helmholtz equation. This scheme development involves relating the derivative terms, namely, uxx and uxxxx, to u at the two adjacent nodal points in order to yield a 3-point stencil implicit scheme. In addition to sixth-order accuracy, the proposed scheme can conditionally provide oscillation-free solutions using the underlying M-matrix theory. Computations have been carried out for several problems which are amenable to exact solutions. Agreement with the exact solutions is excellent even in a grid system involving fewer number of mesh points. Thus, the integrity of the compact scheme and the efficiency of the alternating direction implicit algorithm are demonstrated. The proposed scheme is applied to study the effect of variable wave number on the wave propagation characteristics for the problem, which is partly bounded by a scatter surface.

Marching schemes for inverse acoustic scattering problems

For the numerical solution of inverse Helmholtz problems the boundary value problem for a Helmholtz equation with spatially variable wave number has to be solved repeatedly. For large wave numbers this is a challenge. In the paper we reformulate the inverse problem as an initial value problem, and describe a marching scheme for the numerical computation that needs only n2 log n operations on an n × n grid. We derive stability and error estimates for the marching scheme. We show that the marching solution is close to the low-pass filtered true solution. We present numerical examples that demonstrate the efficacy of the marching scheme.