Finite difference method

Abstract: We compute the continuum thermohydrodynamical limit of a new formulation of lattice kinetic equations for thermal compressible flows, recently proposed by Sbragaglia et al. [J. Fluid Mech. 628, 299 (2009)]. We show that the hydrodynamical manifold is given by the correct compressible Fourier–Navier–Stokes equations for a perfect fluid. We validate the numerical algorithm by means of exact results for transition to convection in Rayleigh–Benard compressible systems and against direct comparison with finite-difference schemes. The method is stable and reliable up to temperature jumps between top and bottom walls of the order of 50% the averaged bulk temperature. We use this method to study Rayleigh–Taylor instability for compressible stratified flows and we determine the growth of the mixing layer at changing Atwood numbers up to At∼0.4. We highlight the role played by the adiabatic gradient in stopping the mixing layer growth in the presence of high stratification and we quantify the asymmetric growth rate...

Abstract: An efficient method for analyzing cavity structures by using the fast Fourier transform (FFT)/Pade technique, in combination with the finite-difference time-domain (FDTD) method, is presented. Without sacrificing the accuracy of the results, this new method significantly reduces the computational time compared to that needed where the conventional FFT algorithm is used. The usefulness of this approach is demonstrated by modeling a lossy cavity and computing its resonant frequencies as well as Q.

Abstract: A nonlinear analysis is carried out for the motion of the inviscid, incompressible fluid in a two-dimensional, rigid, open container which is subjected to forced sinusoidal pitching oscillation. Firstly, the problem is defined as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions. Next, the problem is formulated in the form of a pseudo-variational principle, which provides a basis for our discretization. The finite element method and finite difference method are used spacewise and timewise, respectively. Due to the strong nonlinearity of the problem, an incremental method is used for the numerical analysis. Numerical results obtained by the present method are compared with solutions of the linear theory and experimental data. The difference between linear and nonlinear analysis has been clearly indicated.

Abstract: I have developed and solved the constant- Q model for the attenuation of P- and S-waves in the time domain using a new modeling algorithm based on fractional derivatives. The model requires time derivatives of order m+2γ applied to the strain components, where m=0,1,… and γ= (1∕π) tan−1 (1∕Q) , with Q the P-wave or S-wave quality factor. The derivatives are computed with the Grunwald-Letnikov and central-difference fractional approximations, which are extensions of the standard finite-difference operators for derivatives of integer order. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries and general material-property variability. I verified the results by comparison with the 2D analytical solution obtained for wave propagation in homogeneous Pierre Shale. Moreover, the modeling algorithm was used to compute synthetic seismograms in heterogeneous media corresponding to a crosswell seismic experiment.