Splitting Methods for Problems with Different Timescales

A Consistent Hybrid Finite-Volume/Particle Method for the PDF Equations of Turbulent Reactive Flows

A simple barotropic model is integrated for 39 days by one-hour time steps over 94 per cent of the earth's surface. By suitably smoothing the stream function at each time step, time truncation errors are controlled and the total energy of the system varies within only two per cent of the initial value over the entire integration period. By initiating the integration with energy in the mean flow and in planetary waves one and three, a pronounced periodic exchange of energy is observed over the integration period. The predominant periodicity is a six-day exchange between planetary wave number three and the mean flow. The energy distribution in the higher wave numbers appears to be capable of a statistical interpretation.

The performance of improved initial estimates and ‘heuristic’ and ‘adaptive’ techniques for time step control in the iterative solution of Richards equation is evaluated. The so‐called heuristic technique uses the convergence behaviour of the iterative scheme to estimate the next time step whereas the adaptive technique regulates the time step on the basis of an approximation of the local time truncation error. The sample problems used to assess these various schemes are characterized by nonuniform (in time) boundary conditions, sharp gradients in the infiltration fronts, and discontinuous derivatives in the soil hydraulic properties. It is found that higher order initial solution estimates improve the convergence of the iterative scheme for both the heuristic and adaptive techniques, with greater overall performance gains for the heuristic scheme, as could be expected. It is also found that the heuristic technique outperforms the adaptive method under strongly nonlinear conditions. Previously reported observations suggesting that adaptive techniques perform best when accuracy requirements on the numerical solution are very stringent are confirmed. Overall both heuristic and adaptive techniques have their limitations, and a more general or mixed time stepping strategy combining truncation error and convergence criteria is recommended for complex problems. Copyright © 2006 John Wiley & Sons, Ltd.

The grid partial analytical solution method is a newly developed unconditionally stable explicit numerical solution method for solving parabolic partial differential equations. This method discretizes only the spatial domain and predicts a continuous time dependent function at each spatial nodal point. As such, instead of conventionally predicting the solution from a number set, this method predicts from a functional domain. The typical properties of the grid partial analytical solution method can be summarized as the following: (1) It predicts a continuous nodal time dependent function rather than a discrete nodal value. (2) The prediction is unconditionally stable. And unlike any other unconditionally stable finite difference schemes which will lose accuracy when Fourier number becomes large, the proposed method allows single step time marching and unlimited reduction in the spatial step size Δx. (3) For a fixed time step, the higher value of the grid Fourier number resulting from decreasing Δx, the higher the accuracy is achieved in the predicted solution. (4) The grid partial analytical solution converges uniformly to the full analytical solution as the spatial truncation error is infinitely decreased by reducing the spatial step size Δx. This unique characteristic of the analytical treatment of time also makes it possible to treat other time dependent nonhomogeneities involved in heat conduction problem analytically. In this paper, a moving source heat conduction problem is posed and its grid partial analytical solution method developed.

Numerical calculations for a time-accurate solution of the equations of e uid dynamics often require a timestep constraint. One can reduce this constraint to an inequality relating the time step, the grid spacing, and some reference wave velocity. Historically, the literature in numerical analysis refers to this parametric cluster as the Courant number (nondimensional) and the condition for the linear case as the Courant ‐Friedrichs‐Lewy (CFL)condition. Classically, numerical analysis relies on linearization and von Neumann’ s useof Fourier series to derive the CFL condition. In practice, computational e uid dynamics mostly relies on rules of thumb and heuristic arguments to justify the equation that determines time-step size and numerical stability for complicated and nonlinear calculations. The approach proposed in this paper uses the second law of thermodynamics as a way of imposing a restriction on the time step, applied to linear and nonlinear equations and systems of equations like the equations of gas dynamics. Basically, by transforming the truncation error for the numerical formula approximating a conservation equation into an equation representing the balance of entropy, one can obtain an inequality that restricts the time step to satisfy the second law. The second law as developed extends its role by analogy for the simple linear advection equation, then a nonlinear equation, and e nally a system of equations representing the one-dimensional equations of gas dynamics. In each case results obtained agree with the classical approachforlinearequationsbutdifferinothers,indicatingthatthesecondlawhassignie cantimplicationsbeyond its role in thermodynamics. This work develops the topic only for explicit numerical algorithms with truncation errors no greater than second order. By conjecture one expects that the most general conclusions will hold for implicit and higher-order methods because of the universality of the second law and the concept of entropy.

The formulation of multiple-time-step integrators can provide substantial computational savings for mechanical systems with multiple time scales. However, the scope of these savings may be severely limited by the range of allowable time step choices. In this paper we have performed an exhaustive study of the linear stability of the fully asynchronous methods called AVI (asynchronous variational integrator), with two time steps, for essentially any combination of their values. We have obtained approximate analytical expressions for the time step ratios that may render the scheme unstable in the case of linear equations, and verified them with extensive numerical computations. Synchronous multiple time stepping schemes such as r-RESPA show resonances when the outer step is a multiple of the effective half period of one of the fast oscillators. An elegant generalization is derived in the fully asynchronous case of AVI.

An investigation is conducted of the acceleration of the iterative implementation of the two-dimensional infinite impulse response (IIR) filter proposed by D.E. Dudgeon (IEEE Trans. Acoust. Speech Signal Proc., vol.ASSP-28, no.6, p.666-71, 1980). The conventional procedure results in a sequence of estimations that converge linearly to the desired filter output. Applying an accelerated convergence algorithm for linearly distorted signal reconstruction problems, the authors use a product expansion of the denominator of the transfer function of an IIR filter to derive a class of iterative algorithms that have a pth-order rate of convergence. Specifically, k iterations of the proposed pth-order algorithm are equal to the p/sup k/ iterations of the linear algorithm. Therefore, the number of iterations required to obtain a given approximation is reduced from p/sup k/ to k. The authors show that the spatial truncation error does not affect the computation of the desired result in the region of interest, provided that the boundary values outside the area of interest are known. Finally, the authors present two examples in order to compare the convergence rates for the linear and the accelerated algorithms. >

The quasidiffusion (QD) method is an established and efficient iterative technique for solving particle transport problems. Each QD iteration consists of a high-order SN sweep, followed by a low-order QD calculation. QD has two defining characteristics: (1) its iterations converge rapidly for any spatial grid and (2) the converged scalar fluxes from the high-order SN sweep and the low-order QD calculation differ, by spatial truncation errors, from each other and from the scalar flux solution of the SN equations. In this paper, we show that by including a transport consistency factor in the low-order equation, the converged high-order and low-order scalar fluxes become equal to each other and to the converged SN scalar flux. However, the inclusion of the transport consistency factor has a negative impact on the convergence rate. We present numerical results that demonstrate the effect of the transport consistency factor on stability.

Progress in spectral nodal methods applied to discrete ordinates transport problems

A hybrid spectral nodal method for multigroup X,Y-geometry discrete ordinates eigenvalue problems

The scattering source iterative (SI) scheme is traditionally applied to converge fine-mesh numerical solutions to fixed-source discrete ordinates (SN) neutron transport problems. The SI scheme is very simple to implement under a computational viewpoint. However, the SI scheme may show very slow convergence rate, mainly for diffusive media (low absorption) with several mean free paths in extent (low leakage). In this work we describe an acceleration technique based on an improved initial guess for the scattering source distribution within the slab. In other words, we use as initial guess for the fine-mesh scattering source, the coarse-mesh solution of the neutron diffusion equation with special boundary conditions to account for the classical SN prescribed boundary conditions, including vacuum boundary conditions. Therefore, we first implement a spectral nodal method that generates coarse-mesh diffusion solution that is completely free from spatial truncation errors, then we reconstruct this coarse-mesh solution within each spatial cell of the discretization grid, to further yield the initial guess for the fine-mesh scattering source in the first SN transport sweep (forward and backward) across the spatial grid. We consider a number of numerical experiments to illustrate the efficiency of the offered diffusion synthetic acceleration (DSA) technique.

In the present work, the capacity of phase field method to highlight microstructural changes during the spinodal decomposition of a given binary alloy basing on the Cahn-Hilliard equation is presented. Then, growth and coarsening of precipitates are studied using the KKS (Kim-Kim-Suzuki) model, which includes Cahn-Hilliard and Allen-Cahn equations. The implementation of time stepping algorithms to resolve Phase-Field equations is illustrated. Within Fourier space, using semi-implicit spectral method, it has been demonstrated that it allows faster computing than schemes based on finite difference method. First, spinodal decomposition of a given binary alloy under isothermal loading is implemented and three time stepping approaches are applied: constant time stepping, non- iterative and an iterative method. While the non-iterative method is faster than the constant time stepping scheme, the iterative one, although relatively more CPU consuming, can guarantee the convergence of the computing. These methods are combined in an innovative approach tested on 1D, 2D and 3D grids. The effectiveness of the adopted adaptive time-stepping algorithm allows resolving equations in reasonable CPU time. It predicts different physical phenomena, such as phase separation and growth and coarsening of precipitates induced by important interfacial energies.

This issue's Research Spotlights section contains three papers. First, “Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems" extends the theory of canards---a method commonly used for smooth systems with multiple time scales---to the case of piecewise linear systems. In two dimensions, a classical canard is a solution to a smooth system of differential equations such that one variable is moving much faster than the other. The result is a periodic solution with a graph that appears almost discontinuous, with the shape resembling a cartoon duck (canard being French for both duck and a small joke). In this paper, the theory is extended to include systems that are only piecewise smooth, such as for Filippov vector fields. In addition to new results, this paper presents a self-contained introduction to recent approaches in multiple time scale systems. It is worth a read for those interested in systems with multiple time scales, particularly those with an interest in neuronal modeling. The second article, “Communicability Angle and the Spatial Efficiency of Networks," considers communication processes in a connected network, with the goal of understanding whether the graphs are spatially efficient. The paper develops a new measure of spatial efficiency, called the communicability angle, relating this angle to more traditional graph measures, such as the level of clustering and the existence of holes. The methods are applied to a series of 120 real-world data networks, including networks of urban streets, the power grid, the Internet, and the brain. This compelling new idea would be worthwhile for anyone with interest in theoretical, computational, or data networks. The third paper in the section uses a stochastic birth-death population model for competitive growth between a population of healthy cells and a population of cancer cells. “An Evolutionary Model of Tumor Cell Kinetics and the Emergence of Molecular Heterogeneity Driving Gompertzian Growth" explores the formation of tumors under this model and investigates different treatment methods within the subclinical, clinical, and lethal regimes of tumor growth. This paper is an excellent example of the power of mathematical modeling: The model borrows ideas from stochastic population models which were developed in a completely different applied context. It allows for new insights on tumor control, as it is possible to systematize the finding of optimized strategies, and it is also possible to develop therapies for subclinical tumors, which with current methods would not be clinically detected.

We study domain decomposition counterparts of the classical alternating direction implicit (ADI) and fractional step (FS) methods for solving the large linear systems arising from the implicit time stepping of parabolic equations. In the classical ADI and FS methods for parabolic equations, the elliptic operator is split along coordinate axes; they yield tridiagonal linear systems whenever a uniform grid is used and when mixed derivative terms are not present in the differential equation. Unlike coordinate-axes-based splittings, we employ domain decomposition splittings based on a partition of unity. Such splittings are applicable to problems on nonuniform meshes and even when mixed derivative terms are present in the differential equation and they require the solution of one problem on each subdomain per time step, without iteration. However, the truncation error in our proposed method deteriorates with smaller overlap amongst the subdomains unless a smaller time step is chosen. Estimates are presented for the asymptotic truncation error, along with computational results comparing the standard Crank-Nicolson method with the proposed method.

Local and global error estimations in linear structural dynamics