Scalar Test Problem Research Articles

Trees, Stumps, and Applications

The traditional derivation of Runge–Kutta methods is based on the use of the scalar test problem y′(x)=f(x,y(x)). However, above order 4, this gives less restrictive order conditions than those obtained from a vector test problem using a tree-based theory. In this paper, stumps, or incomplete trees, are introduced to explain the discrepancy between the two alternative theories. Atomic stumps can be combined multiplicatively to generate all trees. For the scalar test problem, these quantities commute, and certain sets of trees form isomeric classes. There is a single order condition for each class, whereas for the general vector-based problem, for which commutation of atomic stumps does not occur, there is exactly one order condition for each tree. In the case of order 5, the only nontrivial isomeric class contains two trees, and the number of order conditions reduces from 17 to 16 for scalar problems. A method is derived that satisfies the 16 conditions for scalar problems but not the complete set based on 17 trees. Hence, as a practical numerical method, it has order 4 for a general initial value problem, but this increases to order 5 for a scalar problem.

Spectral Deferred Corrections with Fast-wave Slow-wave Splitting

The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves (``fast-wave slow-wave problem''). We show that for a scalar test problem with two imaginary eigenvalues $i \lambda_{\text{f}}$, $i \lambda_{\text{s}}$, having $\Delta t ( | \lambda_{\text{f}} | + | \lambda_{\text{s}} | ) < 1$ is sufficient for the fast-wave slow-wave SDC (fwsw-SDC) iteration to converge and that in the limit of infinitely fast waves the convergence rate of the nonsplit version is retained. Stability function and discrete dispersion relation are derived and show that the method is stable for essentially arbitrary fast-wave CFL numbers as long as the slow dynamics are resolved. The method causes little numerical diffusion and its semidiscrete phase speed is accurate also for large wave number modes. Performance is studied for an acoustic-advection problem and for the linearised Boussinesq equations, describing compressible, stratified flow. fwsw-SDC is compared to diagonally implicit Runge--Kutta (DIRK) and implicit-explicit (IMEX) Runge--Kutta methods and found to be competitive in terms of both accuracy and cost.

A step-size selection strategy for explicit Runge–Kutta methods based on Lyapunov exponent theory

Using a stability result for variable time-step Runge–Kutta methods applied to nonautonomous real linear scalar test problem that decays exponentially fast, a step-size selection algorithm is devised. The step-size selection algorithm is based partly on stability information obtained by estimating the discrete Lyapunov exponent of a Runge–Kutta method applied to a nonautonomous linear scalar problem. The utility of the approach is illustrated in numerical experiments that demonstrate how the new algorithm performs against a standard accuracy based step-size selection strategy.

Comparison of multigrid algorithms for high‐order continuous finite element discretizations

SummaryWe present a comparison of different multigrid approaches for the solution of systems arising from high‐order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev‐accelerated Jacobi, and the symmetric successive over‐relaxation smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: (1) high‐order h‐multigrid, which uses high‐order interpolation and restriction between geometrically coarsened meshes; (2) p‐multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different‐order basis functions; and (3) a first‐order approximation multigrid preconditioner constructed using the nodes of the high‐order discretization. This latter approach is often combined with algebraic multigrid for the low‐order operator and is attractive for high‐order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both h‐multigrid and p‐multigrid work well; the first‐order approximation is less efficient. For constant coefficients, all smoothers work well. For variable coefficients, Chebyshev and symmetric successive over‐relaxation smoothing outperform Jacobi smoothing. While all of the tested methods converge in a mesh‐independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared with using multigrid as a solver. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Reciprocal polynomial extrapolation vs Richardson extrapolation for singular perturbed boundary problems

The reciprocal polynomial extrapolation was introduced in Amat et al. (J Comput Math 22(1):1–10, 2004), where its accuracy and stability were studied and a linear scalar test problem was analyzed numerically. In the present work, a new step in the implementation of the reciprocal polynomial extrapolation, ensuring at least the same behavior as the Richardson extrapolation, is proposed. Looking at the reciprocal extrapolation as a Richardson extrapolation where the original data is nonlinearly modified, the improvements that we will obtain should be justified. Several theoretical analysis of the new extrapolation, including local error estimates and stability properties, are presented. A comparison between the two extrapolation techniques is performed for solving some boundary problems with perturbation controlled by a small parameter ϵ. Using two specific boundary problems, the error and the robustness of the new technique using centered divided differences in a uniform mesh are investigated numerically. They turn out to be better than those presented by the Richardson extrapolation. Finally, investigations on the accuracy when using a special non-uniform discretization mesh are presented. A numerical comparison with the Richardson extrapolation for this particular case, where we present some improvements, is also performed.

Fourier Analysis of GMRES(m) Preconditioned by Multigrid

This paper deals with convergence estimates of GMRES(m) [Saad and Schultz, { SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869] preconditioned by multigrid [Brandt, Math. Comp., 31 (1977), pp. 333--390], [Hackbusch, Multi-Grid Methods and Applications, Springer, Berlin, 1985]. Fourier analysis is a well-known and useful tool in the multigrid community for the prediction of two-grid convergence rates [Brandt, Math. Comp., 31 (1977), pp. 333--390], [Stuben and Trottenberg, in Multigrid Methods, Lecture Notes in Math. 960, K. Stuben and U. Trottenberg, eds., Springer, Berlin, pp. 1--176]. This analysis is generalized here to the situation in which multigrid is a preconditioner, since it is possible to obtain the whole spectrum of the two-grid iteration matrix. A preconditioned Krylov subspace acceleration method like GMRES(m) implicitly builds up a minimal residual polynomial. The determination of the polynomial coefficients is easily possible and can be done explicitly since, from Fourier analysis, a simple block-diagonal two-grid iteration matrix results. Based on the GMRES(m) polynomial, sharp theoretical convergence estimates can be obtained which are compared with estimates based on the spectrum of the iteration matrix. Several numerical scalar test problems are computed in order to validate the theoretical predictions.

Dynamic refinement algorithms for spectral element methods

A numerical method is presented that combines high-order domain decomposition and adaptive mesh refinement. The spatial discretization is based on a nonconforming spectral element method. Meshes are generated by adaptive refinement of a quadtree using either local solution gradients or the local polynomial spectrum as a refinement criteria. The method is applied to solve scalar test problems and to simulate the flow in a lid-driven cavity at moderate Reynolds numbers.

Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.

MULTIDIMENSIONAL SINGLE-STEP VECTOR UPWIND SCHEMES FOR HIGHLY CONVECTIVE TRANSPORT PROBLEMS

Abstract After a synthesis of existing numerical approaches, a family of single-step time-marching upwind schemes is proposed for the convective-diffusive balance of a scalar in multidimensional incompressible laminar flows. A functional description of the new approach is given too. The schemes are adequate to the simulation of purely or highly convective transport phenomena. The proposed schemes were applied to a scalar test problem (rotating hill) and the results evaluated after an analysis of the intrinsic limitations of test itself, A third-order, fully upwind-biased scheme of the family is applied to the lid-driven and thermally driven square cavity problems. In the lid-driven case a steady-state solution is not achieved at Re = 104.