Highly accurate pseudospectral approximations of the. A pseudospectral method for nonlinear wave hydrodynamics wooyoung choi and christopher kent university of michigan, usa abstract we present a new hybrid asymptoticnumerical method to study nonlinear wavebody interaction in threedimensional water of arbitrary depth. Stability of a chebychev pseudospectral solution of the. Fourthorder rungekutta scheme is applied for temporal discretization. In this proposal, the fourier pseudospectral timedomain is used, because of its reduced computational cost compared to other timedomain classical. Analysis of twodimensional photonic crystals using a multidomain pseudospectral method pojui chiang,1 chinping yu,1 and hungchun chang1,2,3, 1graduate institute of electrooptical engineering, national taiwan university, taipei, taiwan 10617, republic of china 2graduate institute of communication engineering, national taiwan university, taipei, taiwan 10617. Implementation of pseudospectral method for the solution of the wave equation in acoustic media and comparison with finitedifference method andrea lopez, francisco h. In order to reduce the influence of different numerical methods, we try to use a uniform method pseudospectral method to solve the wave equations.
We have called this method the lanczoschebyshev pseudospectral lcps method. In addition, an application of spectral and pseudospectral approximation to dispersive nonlinear wave equation, such as kdv equation has attracted a great deal of attention. Towards that end, this paper is devoted to a new proposed method based on a damped wave equation and a spatialdependent damping parameter profile to reduce the wave energy on the nearby boundaries. Multisymplectic fourier pseudospectral method for the kawahara equation volume 16 issue 1 yuezheng gong, jiaxiang cai, yushun wang skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. A pseudospectral chebychev method for the 2d wave equation with domain stretching and absorbing boundary conditions. Pseudospectral methods for wave equations on spherical. The generalized stability and the convergence are proved. Introduction the pseudospectral method in a nutshell. Treeby1, 1department of medical physics and biomedical engineering, university college london, united kingdom.
The optimization problem was solved using a pseudospectral method, a direct optimal control approach that can incorporate practical design constraints, such as power. Stability of gaussradau pseudospectral approximations of. It is better to solve equation in the frequency domain because of the frequencydependent parameter v. Pdf a pseudospectral chebychev method for the 2d wave. It combines pseudospectral ps theory with optimal control theory to produce ps optimal control theory. The proposed chebyshev pseudospectral method yields an exponential rate of convergence when the solution is smooth and allows a great flexibility to simultaneously handle fractional time and space derivatives with different levels of truncation. Absorbing boundaries for pseudospectral methods 325 acoustic wave equation modi. In this paper, we generalize this method to twodimensional vorticity equations. Multisymplectic fourier pseudospectral method for the. Analysis of twodimensional photonic crystals using a. The techniques have been extensively used to solve a. The pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling. We examine spectral and pseudospectral methods as well as waveform relaxation methods for the wave equation in one space dimension.
Fourier pseudospectral solution of the regularised long. Chebyshev pseudospectral method for wave equation with. A pseudospectral method for nonlinear wave hydrodynamics. The use of the fourier collocation spectral method in particular has many computational advantages, including a reduced number of grid points required for. A pseudospectral procedure for the solution of nonlinear wave. Pdf numerical solution for the fractional wave equation using. Pseudospectralelement modelling of elastic waves in matlab. The numerical method is based on a deuflhardtype exponential wave integrator for temporal integration and the fourier pseudospectral method for spatial discretizations. We first extend the stability analysis of pseudospectral approximations of the onedimensional oneway wave equation \\frac\partial u\partial x cx\frac\partial u\partial x\ given in 11 to general gaussradau collocation methods. Ps optimal control theory has been used in ground and flight systems in military and industrial applications. Highly accurate pseudospectral approximations of the prolate spheroidal wave equation for any bandwidth parameter and zonal wavenumber.
However, equations and cannot be solved by the method. Diagonally implicit multistage integration methods for. In this paper we propose and analyse a pseudospectral solution of the acoustic wave equation with absorbing boundary conditions abcs for the rst order hyperbolic system formulation of the problem. A pseudospectral method for gravitational wave collapse david hilditch, 1andreas weyhausen, and bernd brugmann 1friedrichschilleruniversit at jena, 07743 jena, germany we present a new pseudospectral code, bamps, for numerical relativity written with the evolution of collapsing gravitational waves in mind. Rbfps method and fourier pseudospectral method for. The space derivatives are calculated in the wavenumber domain by multiplication of the spectrum with. Although, the method of 2 yields highly accurate results for small bandwidth parameters, it becomes useless when c. Comparisons with finite differences for the elastic wave equation. Bojarski and others 19, 20, 2530 applied similar ideas to linear scalar wave equations, with applications in acoustics and. In 17,16 stability restrictions were determined through. Pseudospectral timedomain pstd methods for the wave equation. The model equations under consideration are from the boussinesq hierarchy of equations, and allow for appropriate modeling of dispersive shortwave phenomena by including weakly nonhydrostatic corrections to the hydrostatic pressure in the shallow.
In case of, for example, navierstokes turbulence the situation turns. The detailed update equation for the real part of the wave function at thelth stagecanbewrittenas. Spectral versus pseudospectral solutions of the wave. Timesplitting combined with exponential wave integrator. An exponential wave integrator fourier pseudospectral. Pseudospectral timedomain pstd methods for the wave. Implementation of pseudospectral method for the solution. Pseudospectral time domain pstd methods are widely used in many branches of acoustics for the numerical solution of the wave equation, including biomedical ultrasound and seismology.
Comparisons of viscoacoustic wave equations journal of. The benefit of this is that we can get our hands on interesting solutions of complicated wave equations with relatively little code and with only the computing power in your laptop. A note on stability of pseudospectral methods for wave. On the other hand, the authors 7,10 developed a pseudospectral method by using riesz spherical means to get better results. A global fourier pseudospectral method is presented and used to solve a dispersive model of shallow water wave motions. A pseudospectral chebychev method for the 2d wave equation. Symplectic pseudospectral timedomain scheme for solving. Pseudospectral methods are best suited to simple geometries, and in this short course well only consider periodic cartesian domains.
A fourier pseudospectral method for the good boussinesq. A pseudospectral chebychev method for the 2d wave equation with domain stretching and absorbing boundary conditions article pdf available february 1970. The numerical results show the advantage of such a method. Introduction the pseudospectral method in a nutshell the pseudospectral method in a nutshell the pseudospectral method is. Many efforts have been made to develop numerical method for solving this equation, such as the variational iteration method, the finite difference method 610, the fourier pseudospectral method 11, 12, the galerkin element method, 14, the adomian decomposition method 15, 16, the collocation method 17, 18, and others. Renaut department of mathematics, arizona state university, tempe, az 85287, usa abstract diagonally implicit multistage integration methods are employed for the numerical integration in time of. We give asufficient condition on the collocation points for stability whichshows that classical gaussradau. A pseudospectral method for gravitational wave collapse. The fourier method can be considered as the limit of the finitedifference method as the length of the operator tends to the number of points along a particular dimension. Diagonally implicit multistage integration methods for pseudospectral solutions of the wave equation z. Then the system 7 can be solved with any algorithm for integrating ordinary differential equations. A chebyshev pseudospectral method to solve the spacetime.
The lanczoschebyshev pseudospectral method for solution. The fourier pseudospectral method has been verified as an accurate and effective. Theresultsprovide earlystage guidelines for wec design. An algorithm is presented for the fourier pseudospectral solution of the regularised long wave rlw equation. Chebyshev pseudospectral method for wave equation with absorbing boundary conditions that does not use a. Fourier pseudospectral solution for a 2d nonlinear. It has been realized that, among the mentioned spectral approximations, the fourier pseduospectral is most suited method to solve the envelopeequation. Upload 2d navierstokes in stream function formulation project.
Parallel 3d pseudospectral simulation of seismic wave. The method is introduced here in two different ways. Numerical solution for the fractional wave equation using pseudospectral method based on the generalized laguerre polynomials. A secondorder fourier pseudospectral method for the. Our goal is to study block gaussjacobi waveform relaxation schemes which can be efficiently implemented in a. Realising boundary conditions with discrete sine and cosine transforms elliott s. The numerical results indicate that rbfps method can be more accurate than standard fourier pseudospectral method for many nonlinear wave equations. The analysis and solution of wave equations with absorbing boundary conditions by using a related first order hyperbolic system has become increasingly. The symplectic integrators can satisfy the timereversible or symmetric condition 19,20. There the effectiveness of the method, in particular when a stretching transformation 8 is applied to the.