JCO Oncology Advances

Brown K.S.

A technique is described for using Newton's identities to compute the coefficients of multi-step recurrence formulae for simulating ordinary linear differential equations of high order. This approach gives the exact root-matched recurrence formula without having to determine the complex roots of the characteristic polynomials.

Johnson R.W., MacKinnon R.J.

A recent article by Leonard and Mokhtari (1990) discusses the order of accuracy of discrete equations used for numerical simulation. They maintain that discrete operators derived for the finite-volume method should be Taylor-series-analysed using the various points that appear in a finite-volume discrete equation. The current paper argues that any discrete scheme must be analysed from a single point. It is shown that while Leonard's finite-difference QUICK scheme is O(h3)-accurate, his implementation of QUICK for the finite-volume method is only O(h2). An alternative finite-volume formulation is shown to yield a locally third-order-accurate operator in the final discrete equation. Also, third-order-accurate formulae are discussed for various numerical implementations. Numerical examples are included to demonstrate actual rates of convergence for various operators.

MacKinnon R.J., Johnson R.W., Langerman M.A.

We formulate a higher-order (superconvergent) Petrov–Galerkin method by determining, using a finite-difference approximation, the optimal selection of quadratic and cubic modifications to the standard linear test function for bilinear elements. Application of this method to linear elliptic problems results in improved accuracy and higher rates of convergence for problems with constant coefficients and improved accuracy for problems with variable coefficients. Supporting numerical examples are given.

Esikova N.B., Iliev O.P., Vabishchevich P.N.

Heat and mass transfer during crystallization in a square cavity, involving natural laminary convection of liquid metal with temperature-dependent viscosity, are investigated numerically on a grid with 51 × 51 nodes. A brief survey of existing techniques for convection/diffusion phase change problems is given. A fixed grid numerical methodology, based on the fictitious regions method, is used for solving the 2-D Stefan problem coupled with the Navier–Stokes equations in the Boussinesq approximation. The viscosity variation is modelled by an exponential form, ν/ν0 = exp ( – α(T – 1)). The influence of the Grashof number and of the parameter α on heat and mass transfer are investigated.

Liu Z., Chen S.

Raymond W.H.

A numerical procedure is used to identify the relationship between recursive and implicit filters. We find that the implicit filters can each be decomposed into two recursive filters at each grid point location by using Choleski matrix factorization. Specifically, the implicit second-order sine filter yields the traditional first-order recursive filter, while we recover the standard second-order recursive sine filter from its fourth-order implicit counterpart. This approach thus provides an alternative to traditional methods for calculating the coefficients associated with the recursive sine filters. We also find that we cannot recover the traditional form for the recursive tangent filter from the implicit tangent filter.

Anturkar N.R., Papanastasiou T.C., Wilkes J.O.

SUMMARY An algorithm based on a compound matrix method is presented for solving difficult eigenvalue problems of n equation sets in connected domains that are coupled through (n - 1) sets of interfacial boundary conditions, when n is an arbitrary number. As an example, a linear stability problem of n-layer plane Poiseuille flow is formulated. The resulting Orr-Sommerfeld equations form a set of stiff differential equations at high wavenumbers, which are solved accurately for various combinations of parameters. and Davey' for two-point boundary value problems and eigenvalue problems of the Orr-Sommerfeld equation. These investigations computed eigenvalues and eigenfunctions with marginal errors where standard shooting methods failed. Yiantsios and Higgins6 extended the method to equation sets valid over two connected domains that are coupled through interfacial conditions; in particular, they solved the Orr-Sommerfeld equations for two superposed fluids in plane Poiseuille flow. However, since the eigenvalues are obtained by matching the interfacial boundary conditions, the method is unsuitable for computations in more than two domains connected by interfacial boundary conditions. Here, the compound matrix method is implemented for n equation sets that are valid over connected domains through (n - 1) sets of interfacial boundary conditions, where n is arbitrary. Instead of matching boundary conditions at a particular interface for finding eigenvalues, the integration of the compound differential system proceeds with new initial conditions at an interface for the compound differential system of the next domain. Subsequently, the eigenvalues are computed by matching the boundary conditions at the end of the last domain. This general algorithm for finding eigenpairs is used in the linear stability analysis of n-layer, Newtonian, plane Poiseuille flow, for which resulting differential systems are known to be stiff at large wavenumbers.

Moussa N.

This work is based on the constructive existence proof of solutions of a comprehensive class of non-linear free boundary-value problems of plane hydrodynamics by E. Zeidler (1971) and a general computational method for constructing the solutions numerically given by the author (1991). The case of permanent capillar-heavy waves in infinite channels will be considered.

Cai Z., Weinan E.

In this paper we explore the hierarchical structures of wavelets, and use them for solving the linear systems which arise in the discretization of the wavelet–Galerkin method for elliptic problems. It is proved that the condition number of the stiffness matrix with respect to the wavelet bases grows like O(log2H/h) in two dimensions, and the condition number of the wavelet preconditioning system is bounded by O(log2H/h) in d dimensions, instead of O(h−2) if the scaling function bases are used.

Catalano L.A., Napolitano M., Deconinck H.

An optimal multistage explicit scheme is applied to solve a non-linear advection equation using a standard first-order-accurate upwind scheme and three different genuinely two-dimensional first-order-accurate discrete operators, within a finite-volume context. Such an explicit time integration procedure is combined with a classical multigrid method to accelerate convergence to steady state.

Rossow C.-.

The efficient implementation of a multigrid technique for use on block-structured, body-fitted meshes is investigated. The algorithm employed for the solution of inviscid flow fields uses central differences with artificial dissipation, and time integration is performed by a five-stage Runge—Kutta scheme. A major point of concern is the amount of work which may be performed within a block before switching to the next block. In this study three different strategies are under consideration. In the first strategy data are exchanged between blocks in each stage of the Runge—Kutta time-stepping scheme. This keeps a possible time lag between blocks to a minimum, but requires a large amount of I/O operations and storage. The second strategy performs a complete Runge—Kutta cycle within a block before switching to the next. Having completed one time step in all blocks, a second sweep through all blocks is done in order to evaluate residuals necessary for the restriction to coarser meshes. In the third strategy both a complete Runge—Kutta cycle and the residual evaluation for the restriction operator are done within a block, allowing a minimum of I/O and storage. A primary application of all three strategies to a two-block mesh around a wing-body combination showed no significant differences in the convergence of the multigrid algorithm. To investigate the impact of the different strategies on complex configurations, computations were carried out on an 11-block mesh around a wing-body—engine-pylon configuration. Both the first and second strategies delivered converged results, but the third failed due to the larger time lag between blocks. Full multigrid was then used to obtain a preconditioned starting solution for the finest grid. This alleviated the time lag to such an extent that even the third strategy yielded convergence.

Fulton S.R.

Multigrid algorithms are presented for several elliptic problems arising in models of balanced atmospheric dynamics. The equations are generalizations of the Monge—Ampere equation and are non-linear in the highest-order terms. The problem formulation is reviewed, along with a method of dealing with a lower boundary which is not a co-ordinate surface, and details of the discretization and relaxation are given. Numerical results from two- and three-dimensional problems show the accuracy and efficiency of the method.