Pseudospectral Research Articles

Pseudospectral Solution of Near-Singular Problems using Numerical Coordinate Transformations Based on Adaptivity

The work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions. The approach makes use of the adaptive finite difference method presented by Huang and Sloan [SIAM J. Sci. Comput., 15 (1994), pp. 776--797]. This method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh. The adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space. The map between the spaces is based on Chebyshev polynomial interpolation. Finally, the standard pseudospectral (PS) method is applied to the transformed differential problem to obtain highly accurate, nonoscillatory numerical solutions. Numerical results are presented for steady problems in one and two space dimensions.

Comparison of finite difference‐ and pseudospectral methods for convective flow over a sphere

For modeling convective flows over a sphere or within a spherical shell, pseudospectral (PS) methods are in general far more cost‐effective than finite difference‐ (or finite element) methods. This study confirms this, and proceeds by comparing a Fourier‐PS implementation (based on a longitude‐latitude grid) with one using spherical harmonics. For similar resolutions, these are found to be of similar accuracy. However, the former is computationally about ten times faster.

Pseudospectral localized generalized Mo/ller–Plesset methods with a generalized valence bond reference wave function: Theory and calculation of conformational energies

We describe a new multireference perturbation algorithm for ab initio electronic structure calculations, based on a generalized valence bond (GVB) reference system, a local version of second-order Mo/ller–Plesset perturbation theory (LMP2), and pseudospectral (PS) numerical methods. This PS-GVB-LMP2 algorithm is shown to have a computational scaling of approximately N3 with basis set size N, and is readily applicable to medium to large size molecules using workstations with relatively modest memory and disk storage. Furthermore, the PS-GVB-LMP2 method is applicable to an arbitrary molecule in an automated fashion (although specific protocols for resonance interactions must be incorporated) and hence constitutes a well-defined model chemistry, in contrast to some alternative multireference methodologies. A calculation on the alanine dipeptide using the cc-pVTZ(−f) basis set (338 basis functions total) is presented as an example. We then apply the method to the calculation of 36 conformational energy differences assembled by Halgren and co-workers [J. Comput. Chem. 16, 1483 (1995)], where we obtain uniformly good agreement (better than 0.4 kcal/mole) between theory and experiment for all test cases but one, for which it appears as though the experimental measurement is less accurate than the theory. In contrast, quadratic configuration interaction QCISD(T) calculations are, surprisingly, shown to fail badly on one test case, methyl vinyl ether, for which the calculated energy difference is 2.5 kcal/mole and the experimental value is 1.15 kcal/mole. We hypothesize that single reference methods sometimes have difficulties describing multireference character due to low lying excited states in carbon–carbon pi bonds.

Local quantum chemistry. Incorporation of pseudospectral methodology into the local space approximation

We show that introducing pseudospectral (PS) methodology into local space approximation (LSA) treatments reduces the scaling with system size by a factor of N, N being a measure of size, at both Hartree-Fock and correlated levels. Efficient ways to organize Hartree-Fock calculations, avoiding storage of full and local space two-electron integrals, are presented. A similar analysis applies if the PS approach is replaced by so-called resolution of the identity integral evaluation. The possibility of major savings of computer time, unique to LSA calculations and due to the rapid fall-off of density matrix elements with interatomic distance, is discussed.

Axisymmetrization of an isolated vortex region by splitting and partial merging of satellite depletion perturbations

In numerical studies, we observe the essentially inviscid approach to axisymmetry of an isolated-and-perturbed monopolar and uniformly decreasing distributed vortex region in a two-dimensional incompressible fluid. In particular, an initial small-but-finite amplitude, 3-fold, ‘‘edge’’-located hole or depletion perturbation of a monopole evolves into a 2-fold state. This occurs through the nonlinear processes of hole stretching, splitting and partial merger. The initial growth rate for this downward cascade is proportional to the initial perturbation magnitude. Near-inviscid simulations are made with surgery-regularized contour dynamics codes (CDS), using from 5 to 11 contours. Pseudospectral (PS) simulations with varying Newtonian and hyperviscosities, that is considering continuum and dissipation effects, yield consistent results. Our numerical results are in agreement with small-but-finite amplitude perturbations that were used in recent laboratory experiments on magnetized electron columns. This process may also occur in late time evolutions associated with the ‘‘bump-on-tail’’ initial condition in Vlasov plasmas.

Cost‐effective staggered schemes for the numerical simulation of wave propagation

AbstractThis paper presents a cost‐effectiveness analysis of explicit Finite Difference (FD) methods for the numerical integration of the wave equation. Formal notions of computational cost (expressed in floating point operations) and numerical dispersion error are introduced. Restricting the analysis to leapfrog timemarching, for sake of simplicity, various spatial discrete differentiators are examined. For each scheme, by minimizing the cost at a given error threshold, a cost‐effective operating poin (time sampling rate and number of gridpoints per shortest wavelength) is obtained, which is remarkably different from the stability limit. Different schemes, each operated at its cost‐effective point, are then compared. High‐order dispersion‐bounded operators, in the sense of Holberg,1 are found to be competitive with the Pseudo‐spectral (PS) method.New optimal schemes improving over the Holberg's spatial differentiators are introduced together with accurate expansions of the convolutional weights is terms of the design error threshold. It is also shown that the composition of two distinct Holberg's operators of consecutive orders, with opposite phase properties, minimizes dispersion and yields cost‐effective schemes.Numerical experiments illustrate the suitability of the new methods for large‐scale wave‐equation seismic modelling.

High gradient phenomena in two-dimensional vortex interactions

Hyperdiffusion, a simple linear eddy diffusivity scheme, is commonly used in atmospheric and oceanic simulations because it increases the range of inertially behaving spatial scales for a given model resolution. Compared with molecular diffusion (which is utterly negligible in the atmosphere and oceans), hyperdiffusion more sharply confines the dissipation to the smallest scales of the numerical model. But is this all that hyperdiffusion does? In this paper, the inelastic interaction of two distributed vortices of unequal size is examined. Contour surgery (CS) simulations are compared with pseudospectral (PS) simulations employing hyperdiffusion or molecular diffusion. The example illustrates what is believed to be the most fundamental characteristic of two-dimensional (2-D) (and layerwise-2-D) vortex dynamics, namely, the formation of exceedingly high vorticity gradients. There is an excellent agreement between the hyperdiffusive PS and CS calculations at early times (i.e., for a few vortex rotation periods). Thereafter, significant discrepancies develop, beginning abruptly from the time when vorticity-gradient intensification is arrested by diffusion. A rapid inward erosion of the smaller of the two vortices then takes place. This erosion takes place under the joint action of (hyper) diffusion and stripping (the peeling of the vortex periphery by the external flow). With hyperdiffusion, the erosion is accompanied by a serious numerical artifact: a climb in the peak vorticity by 30% in this example. Eventually, the erosion reaches the vortex center and the vortex is sheared into a filament. In the CS calculation, there is no erosion, no climb in peak vorticity, and the vortex appears to last indefinitely. In the PS calculations, the viscosity or hyperdiffusion is adjusted according to the resolution to give the largest possible inertial range while ensuring numerical stability. It is found that vortices that are spanned by fewer than 10–20 grid points are eroded away in only a few vortex rotation periods (a time scale that is very much shorter than one would estimate from pure viscous decay). These findings bring into question the results of many 2- turbulence simulations using hyperdiffusion, for hyperdiffusion simulates neither inviscid dynamics nor molecular-diffusive dynamics.

Pseudospectral generalized valence-bond calculations: Application to methylene, ethylene, and silylene

The pseudospectral (PS) method for self-consistent-field calculations is extended for use in generalized valence-bond calculations and is used to calculate singlet–triplet excitation energies in methylene, silylene, and ethylene molecules and bond dissociation and twisting energies in ethylene. We find that the PS calculations lead to an accuracy in total energies of ≤0.1 kcal/mol and excitation energies to ≤0.01 kcal/mol for all systems. With effective core potentials on Si, we find greatly improved accuracy for PS.

Dispersion-bounded numerical integration of the elastodynamic equations with cost-effective staggered schemes

Known procedures for designing numerical schemes for the integration of elastodynamic equations with explicit control over numerical dispersion are reviewed. In the literature, the analysis of such schemes has concentrated on the discrete space differentiators, and has neglected the role played by time discretization in the overall accuracy. In this paper we define a computational cost for a given dispersion error bound which fully includes the effect of temporal differencing. For some representative schemes based on leap-frog time marching, we provide an optimal operating point (time sampling rate and number of grid points per shortest wavelength) which minimizes the computational cost for a given dispersion error threshold. Based on this notion of cost, we introduce new optimal operators for staggered grids. Additionally, we introduce the notion of composite differentiators to design still more cost-effective schemes. The cost of the proposed schemes is shown to be less than that of known finite difference (FD) operators and compares favorably with pseudo-spectral (PS) algorithms. Numerical simulations are presented to illustrate the effectiveness of the new operators.