Cross-derivative Terms Research Articles

On the Degradation of Cell-Centered Diffusive Preconditioners for Accelerating SN Transport Calculations in the Periodic Horizontal Interface Configuration

We investigate the degraded effectiveness of diffusion-based acceleration schemes in terms of the adjacent-cell preconditioner (AP) in a periodically heterogeneous limit devised for the two-dimensional (2-D) periodic horizontal interface (PHI) configuration. Specifically, we demonstrate that the diffusive low-order operator employed in the AP scheme lacks the structure of the integral transport operator in the above asymptotic limit since it (1) ignores cross-derivative coupling and (2) incorrectly estimates the strength of intra-layer coupling in the optically thin layers. In order to prove propositions 1 and 2, we derive expressions for the elements of the matrix representing a certain angular (SN) and spatially discretized form of the 2-D neutron transport integral operator. This is the transport operator that produces the full scalar flux solution if it is directly inverted on the once-collided particle source. The properties of this operator’s elements are then investigated in the asymptotic limit for PHI. The results of the asymptotic analysis point to a sparse but nonlocal matrix structure due to long-range coupling of a cell’s average flux with its neighboring cells, independent of the distance between the cells in the spatial mesh. In particular, for a cell in a thin layer, cross-derivative coupling of the cell’s flux to its diagonal neighbors is of the same asymptotic order as self-coupling and coupling with its north/south Cartesian neighbors. Similarly, its coupling with the fluxes in the same thin layer is of the same order, independent of the distance between the cells in the layer, as the coupling with the east/west Cartesian neighbors.We also show that modifying the standard diffusion-based AP can lead to effective acceleration in PHI. Specifically, we devise three novel acceleration schemes, named APB, Optimized-AP (OAP), and Hybrid-AP (HAP), obtained by modifying the original AP formalism in 2-D. In the APB the five-point AP operator is extended to a nine-point stencil that accounts for cross-derivative coupling by including the matrix elements of the integral transport operator B, which couple a cell-averaged scalar flux to its first diagonal neighbors. In the OAP the five-point stencil of the original AP operator is retained while optimizing the value of the elements in the preconditioner that affect the coupling of a cell with its east/west Cartesian neighbors. Specifically, the optimum elements are obtained by minimizing the iteration’s spectral radius and offer a more correct estimate of the strength of intra-layer coupling in a thin layer. Finally, the nine-point HAP operator represents a “hybrid” of the APB and OAP approaches, in the sense that the spectral properties of the optimized five-point OAP are further improved via the inclusion of cross-derivative terms. Fourier analysis of the novel acceleration schemes indicates that robustness of the accelerated iterations can be recovered, in spite of sharp material discontinuities, by accounting for cross-derivative coupling and by optimizing the preconditioner elements. The new acceleration schemes have also been implemented in a 2-D transport code, and numerical tests successfully verify the predictions of the Fourier analysis. However, it is important to emphasize that the modifications attempted in this work are specific to the selected asymptotic limit for PHI and do not translate into new low-order operators for the general heterogeneous-material case. Rather, the above modified operators suggest that it may be possible to eventually derive such a general low-order unconditionally robust operator.

Proton transitions in hydrogen bonds of DNA: A first-order perturbation model

Experimental evidence which supports Lowdin's formulation of the microscopic proton-electron code model of genetic information is reviewed. The data are consistent with interstrand transfer of hydrogen bonded protons at G-C base pairs giving rise to mutations in DNA which replicates infrequently. A first-order perturbation model in terms of Fermi's golden rule is presented as a qualitative description of interstrand proton transitions within hydrogen bonds of DNA. The particular reaction considered is given by O…H-NO-H…N. Cross derivative terms in the kinetic energy operator are neglected, and a piecewise-constant rectangularly shaped potential energy surface is employed for the hydrogen bonding system. The full potential is separated such that the components responsible for the reaction are uniquely defined and then “sandwiched” between initial and final states to yield an expression for the proton transition rate. Analysis of this expression provides a “physical picture” of the way energy transfer can occur between different vibrational modes of the hydrogen bond. Increases in the reactivity of newly opened reactive channels and vibrational enhancement effects are discussed in terms of the model with experimental implications given.

ADI‐FDTD method perturbed by the second order cross derivative terms

AbstractA two‐step FDTD method as a compromise of conditional stability and reduced splitting error is formulated and its numerical stability is investigated. It is the perturbed form to the ADI‐FDTD method by the addition of second order cross derivative term. It is validated from the comparison of numerical anisotropy and numerical error over the ADI‐FDTD that numerical performances can be improved by controlling the perturbed term within the stable region of the cross derivative term. © Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1822–1826, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23479

Stability of ADI schemes applied to convection–diffusion equations with mixed derivative terms

We consider Alternating Direction Implicit (ADI) schemes for the numerical solution of initial-boundary value problems for convection–diffusion equations with cross derivative terms. We derive new linear stability results for three ADI schemes that have previously been studied in the literature. These results are subsequently used to show that the ADI schemes under consideration are unconditionally stable when applied to finite difference discretizations of general parabolic two-dimensional convection–diffusion equations. Supporting numerical evidence is included.

Collocation methods for the solution of von-Kármán dynamic non-linear plate systems

The von-Kármán nonlinear, dynamic, partial differential system over rectangular domains is considered, and numerically solved using both the Chebyshev-collocation and Legendre-collocation methods for the spatial discretization and the implicit Newmark- β scheme combined with a non-linear fixed point algorithm for the temporal discretization. As the system is non-linear, involving operators of different orders, different timescales, and may contain initial/boundary incompatible conditions at the domain's corners, it is our aim to highlight some of the difficulties inherent in its numerical treatment using collocation methods. We begin by examining the Chebyshev-collocation scheme considered in [Comput. Meth. Appl. Mech. Engrg. 193 (6–8) (2004) 575]. In that work, filtering was used to stabilize the full von-Kármán system, however the source of the instability was not discussed. We first show that IC/BC incompatibilities are not the culprits and demonstrate empirically that the source of the instability was the numerical treatment of the linear cross-derivative terms found in the in-plane equations. We prove mathematically that although the continuous linear system is well-posed, the Chebyshev-collocation solution of problems involving cross-derivatives and second-order time derivatives is unstable. Instead of adding sufficient filtering to counter the energy growth due to the linear terms, we employ the Legendre-collocation method. We show empirically and prove that the Legendre-collocation scheme for the linear system has the necessary characteristics to remain stable. Nevertheless, the collocation treatment of the non-linear terms still causes the solution to be unstable at long time. We thus return to advocating a filtering solution, but in this case only filtering the non-linear terms to compensate for the inherent aliasing error due to our collocation treatment of these terms. We conclude by comparing the results of filtered Chebyshev-collocation solution, non-linear terms filtered Legendre-collocation solution, and a simplified von-Kármán (solution omitting the inertial terms in the in-plane equations) as presented in (loc. cit).

An improved numerical simulator for multiphase flow in porous media

AbstractIn basin modelling the thermodynamics of a multicomponent multiphase fluid flux are computationally too expensive when derived from an equation of state and the Gibbs equality constraints. In this article we present a novel implicit molar mass formulation technique using binary mixture thermodynamics. The two proposed solution methods, with and without cross derivative terms between components, are based on a preconditioned Newton‐GMRES scheme for each time‐step with analytical computation of the derivatives. These new algorithms reduce significantly the numerical effort for the computation of the molar masses, and we illustrate the behavior of these methods with numerical computations. Copyright © 2004 John Wiley & Sons Ltd.

Spectral Preconditioners for Nonhydrostatic Atmospheric Models

Abstract The elliptic problems in semi-implicit nonhydrostatic atmospheric models are difficult. Typically, they are poorly conditioned, nonseparable, contain cross-derivative terms, and are often nonsymmetric. Here, the resulting linear system is solved using a preconditioned Krylov subspace method—the generalized conjugate residual (GCR) algorithm. A horizontal spectral preconditioner is developed as an alternative to a more standard and much simpler line Jacobi relaxation scheme. However, the efficacy of the spectral preconditioner requires neglecting the cross-derivative terms and homogenization (e.g., averaging) metric coefficients over the computational domain. Because such a compromise causes a substantial departure of the preconditioner from the original elliptic operator, it is not obvious a priori whether it leads to a competitive solver. The robustness of the proposed approach over a broad range of representative meteorological applications is evaluated, in the context of a three-time-level sem...

Study of impinging turbulent jet flows using the isotropic low-Reynolds number and the algebraic stress methods

The performance of two-equation turbulence models (such as the low Reynolds number k–ɛ model of Launder and Sharma 1974) is evaluated versus the algebraic stress model (ASM) (Rodi 1976) for high Reynolds number (Re=9×104) jet flows with strong streamline curvature due to impingement onto a flat plate. The partial differential equations for the conservation of mass and momentum are solved using a finite volume method and the predicted velocities are compared to experimental data by Myszko (1997). The paper demonstrates that in the free-jet region both models over-predict the thickness of the jet. The ASM predicts faster jet growth rate and smaller jet thickness than the low Reynolds number model resulting to closer agreement with the experiments. As a consequence of the better performace of the ASM in the free-jet region, predictions in the wall-jet region showed that despite the use of the logarithmic law-of-the-wall function, the ASM results are closer to the experimental points than the predictions obtained with the two-equation model. However, the rate of peak velocity decay is far higher than the experimental one with both turbulence models. Again, the decay rate predicted with the ASM fits better the exprimental data. The implementation of the ASM exhibited convergence problems most of which were atributed to the cross-derivative terms in the k and ɛ equations and were treated using a linear under-relaxation technique. In general the ASM predictions were more accurate than the low Reynolds number k–ɛ model, with an extra computational cost of less than 25%, which makes the model very attractive for the prediction of turbulence characteristics of high Reynolds number flows with strong stramline curvature.

Pricing Algorithms of Multivariate Path Dependent Options

Financial derivatives which are multivariate in nature are abundant in the financial markets. The underlying state variables may be the stock prices, interest rates, exchange rates, stochastic volatility, average of stock prices, extremum values of stock priors, etc. Option contracts whose life and payoff depend on the stochastic movement of the underlying asset prices are termed path dependent options. In this paper, we examine the pricing methods of several prototype path dependent options. These include options with sequential barriers, options with an external barrier and two-asset lookback options. The governing equations for the option prices are seen to resemble the diffusion type equations but with cross derivative terms, a feature which differs from the usual diffusion equations in engineering. Various techniques to reduce the complexity of the multivariate nature of these prototype option pricing models are discussed. It is illustrated that the dimensionality of a path dependent option model may be reduced by some ingenious choices of similarity variables. We also examine the design of pricing algorithms of these multivariate options, in particular, with regard to the treatment of discrete monitoring feature and the prescription of numerical boundary conditions. The possible generalizations of the numerical techniques presented in this paper to other models with more complicated path dependent payoff structures are also discussed.

Application of the Multigrid Method and a Flexible Hybrid Coordinate in a Nonhydrostatic Model

Abstract A fully compressible, three-dimensional, nonhydrostatic model is developed using a semi-implicit scheme to avoid an extremely small time step. As a result of applying the implicit scheme to high-frequency waves, an elliptic partial differential equation (EPDE) has been introduced. A multigrid solver is applied to solve the EPDEs, which include cross-derivative terms due to terrain-following coordinate transformation. Several experiments have been performed to evaluate the model as well as the performance of the scheme with respect to tolerance number, relaxation choice, sweeps of prerelaxation and postrelaxation, and a flexible hybrid coordinate (FHC). An FHC with two functions (base and deviation functions) is introduced. The basic function provides constant vertical grid spacing required in the multigrid solver, while the deviation function helps to adjust the vertical resolution.

Modified multigrid for 3D elliptic equations with cross-derivatives

A portable Fortran code that solves the general nonseparable three-dimensional linear elliptic partial differential equation (PDE) with cross-derivative terms on a rectangular region is described. Boundary conditions can be any combination of periodic, specified, or mixed derivative. A multigrid scheme, modified to handle complexities introduced by cross-derivative terms and nonnormal derivative components in boundary conditions, is utilized. Successful application is illustrated with an analytically prescribed abstract example and simulation of an idealized geophysical fluid flow.

A Wave Propagation Method for Three-Dimensional Hyperbolic Conservation Laws

A class of wave propagation algorithms for three-dimensional conservation laws and other hyperbolic systems is developed. These unsplit finite-volume methods are based on solving one-dimensional Riemann problems at the cell interfaces and applying flux-limiter functions to suppress oscillations arising from second-derivative terms. Waves emanating from the Riemann problem are further split by solving Riemann problems in the transverse directions to model cross-derivative terms. With proper upwinding, a method that is stable for Courant numbers up to 1 can be developed. The stability theory for three-dimensional algorithms is found to be more subtle than in two dimensions and is studied in detail. In particular we find that some methods which are unconditionally unstable when no limiter is applied are (apparently) stabilized by the limiter function and produce good looking results. Several computations using the Euler equations are presented including blast wave and complex shock/vorticity problems. These algorithms are implemented in the CLAWPACK software, which is freely available.

Full Viscous Modeling in Generalized Coordinates of Heat Conducting Flows in Rotating Systems

A computational and analytical study has been carried out that assesses viscous formulations used currently in the state-of-the-art and widely used incompressible viscous flow solution methodologies in generalized curvilinear coordinates. These methodologies use a powerful and simple approximate factorization delta scheme. The conventional thin (viscous) layer form used for steady-state flows is analyzed and is then improved to extend its applicability, in general, to the solution of unsteady incompressible flows. Then, this improved thin-layer formulation is further extended to include more terms of the same order that are ignored in the thin-layer approximation because of the limitation imposed by the use of a conservation form of the viscous flux vector. This is achieved by recasting the viscous vector in nonconservation form. The resulting formulation represents the full-viscous (FV) formulation (except for higherorder cross-derivative terms). Solutions with the full-viscous, improved, and conventional thin-layer formulations are shown to be different, even for laminar heat conducting flow. Also, the steady state full-viscous solutions with and without the cross-derivative terms are compared and are found to be practically the same. This delineates the need, in general, for the FV formulation for turbulent flows, where turbulent viscosity variation in space is large. The methodology developed here is then used to compute flow in a rotating square duct with fast and slow rotation. The limiting solutions obtained are compared with the analytic limiting solutions and previously obtained reduced numerical solutions. Agreement is shown to be very good.

Preconditioned Conjugate-Residual Solvers for Helmholtz Equations in Nonhydrostatic Models

Numerical integration of the compressible nonhydrostatic equations using semi-implicit techniques is complicated by the need to solve a Helmholtz equation at each time step. The authors present an accurate and efficient technique for solving the Helmholtz equation using a conjugate-residual (CR) algorithm that is accelerated by ADI preconditioners. These preconditioned CR solvers possess four distinct advantages over most other solvers that have been used with the Helmholtz equations that arise in compressible nonhydrostatic semi-implicit atmospheric models: the preconditioned CR methods 1) can solve Helmholtz equations containing variable coefficients, alleviating the need to prescribe a reference state in order to simplify the elliptic problem; 2) transparently include the cross-derivative terms arising from terrain transformations; 3) are efficient and accurate for nonhydrostatic models used across a broad range of scales, from cloud scales to synoptic-global scales; and 4) are easy to formulate and program. These features of the CR solver allow semi-implicit formulations that are unconstrained by the form of the Helmholtz equations, and the authors propose a formulation that is more consistent than those most often used in that it includes implicit treatment of all terms associated with the pressure gradients and divergence. This formulation is stable for nonhydrostatic-scale simulations involving steep terrain, whereas the more common semi-implicit formulation is not. The ADI preconditioners are presented for use in simulations of both hydrostatic and nonhydrostatic scale flows. These simulations demonstrate the efficiency and accuracy of the preconditioned CR method and the overall stability of the model formulation. The simulations also suggest a general convergence criteria for the iterative algorithm in terms of the solution divergence.

Full viscous modeling in generalized coordinates of heat conducting flows in rotating systems

A computational and analytical study has been carried out that assesses viscous formulations used currently in the state-of-the-art and widely used incompressible viscous flow solution methodologies in generalized curvilinear coordinates. These methodologies use a powerful and simple approximate factorization delta scheme. The conventional thin (viscous) layer form used for steady-state flows is analyzed and is then improved to extend its applicability, in general, to the solution of unsteady incompressible flows. Then, this improved thin-layer formulation is further extended to include more terms of the same order that are ignored in the thin-layer approximation because of the limitation imposed by the use of a conservation form of the viscous flux vector. This is achieved by recasting the viscous vector in nonconservation form. The resulting formulation represents the full-viscous (FV) formulation (except for higherorder cross-derivative terms). Solutions with the full-viscous, improved, and conventional thin-layer formulations are shown to be different, even for laminar heat conducting flow. Also, the steady state full-viscous solutions with and without the cross-derivative terms are compared and are found to be practically the same. This delineates the need, in general, for the FV formulation for turbulent flows, where turbulent viscosity variation in space is large. The methodology developed here is then used to compute flow in a rotating square duct with fast and slow rotation. The limiting solutions obtained are compared with the analytic limiting solutions and previously obtained reduced numerical solutions. Agreement is shown to be very good.

Orthogonal Adaptive Grid Generation with Fixed Internal Boundaries for Oil Reservoir Simulation

SUMMARY The generation of boundary conforming, orthogonal, smooth curvilinear grid systems for regions containing internal boundaries, which can also cluster dynamically to physical solution gradients, is of major importance to reservoir simulation. This paper develops a practical methodology for achieving these objectives for two dimensional problems, with extensions to three dimensions. It is based on extending elliptic mapping methods to parabolic ones and the use of equidistribution schemes.

Iterative Line Cubic Spline Collocation Methods for Elliptic Partial Differential Equations in Several Dimensions

This paper presents a new class of second- and fourth-order line cubic spline collocation methods (the LCSC methods) for multidimensional linear elliptic partial differential equations with no cross derivative terms. The LCSC methods approximate the differential operator along lines in each dimension independently and then combine the results into one large linear system. Expressed in terms of discretization stencils for the operator, these methods have nonzero entries only in the coordinate directions. The advantage of this approach is that the discretization is much simpler to derive and analyze. Further, iterative methods are easily applied to the resulting linear systems, especially on parallel computers. The disadvantage is that the resulting linear system is k times larger in k dimensions. Using the simplicity of the methods, iterative schemes are analyzed and formulated in order to solve the resulting LCSC linear systems in the case of Helmholtz problems. Block Jacobi, extrapolated Jacobi (EJ), and successive overrelaxation (SOR) iteration methods are analyzed with the rates of convergence and the optimum relaxation parameters determined. The simple structure of the linear system makes these methods particularly suitable for parallel computation. It is shown that the overall efficiency of the method is attractive in spite of involving such a large linear system. Experimental results presented here confirm the convergence results for both the discretization and iterative methods, and indicate that the convergence results hold for problems more general than Helmholtz problems.

Higher-Order, Finite-Difference Scheme for Three-Dimensional Transonic Flowfields about Axisymmetric Bodies

A higher-order, finite-difference scheme using a third-order shape function is applied to the three-dimensional transonic potential flow problem. Only the firstand cross-derivative terms are more accurately represented than in the conventional second-order accurate schemes. The difference equations are solved by relaxation. Both the convergence rate of the relaxation process and the accuracy for a given number of grid points are improved. The potential equation is solved under an arbitrary, locally defined coordinate transformation, capable of treating general three-dimensional geometries. Numerical results for flowfields about axisymmetric nacelles at angle of attack are presented.

A non-self-adjoint finite difference model of electron diffusion parallel to electric fields

The electron distribution function plays a key role in the quantitative study of various properties of electrons moving through a gas and is characterized as a solution of the Boltzmann transport equation. Simplication of this model results in a non-self-adjoint second-order elliptic partial differential equation with variable coefficients. The complexity of the coefficients precludes the use of a coordinate transformation to remove second-order cross-derivative terms, and the nonmodelness of the boundary value problem necessitates rather novel applications of various tools of numerical analysis for successful solution. Due to the non-self-adjointness, any consistent finite difference model is bound to be nonsymmetric; however, a finite difference scheme is devised which is “almost symmetric.” Iterative solution of the resulting difference equations is analyzed and an acceleration strategy for convergence is devised. Numerical results are included for a typical problem involving in excess of 5000 Mesh points. This problem was solved on the UNIVAC 1106 in less than 6 min of CPU time.

Theory of reactive scattering

Reactive collision probabilities have been calculated in a collinear approximation for a model representing the proton transfer scattering system of Cross derivative terms in the kinetic energy operator are neglected and the potential surface employed is modelled after the ab initio surface calculated by Brown and Hayes. The range of energies studied is 0–3 eV which includes the first four vibrational states of H2 + and the first three states of HeH+. The calculations were performed using the coupled channel τ operator equations of Baer and Kouri. The results are in qualitative agreement with the experimental cross section versus energy data of Chupka, Berkowitz and Russell. The effects of the approximations made are briefly discussed and comparisons with predictions of classical mechanics are made.