Achieving tolerance proportionality in software for stiff initial-value problems

Tolerance proportionality is identified as a minimum requirement for a more uniform interpretation of the user's requested accuracy in software for the solution of initial-value problems. Results are quoted that show that a code must control the local error per unit step in order to achieve tolerance proportionality. This effect can be obtained in codes based on multistep formulas by using a corrector formula of one order higher than the predictor formula. This strategy is simple to implement and leads to better tolerance proportionality than the local error per step strategy often used in stiff software.Numerical results are obtained from two existing solvers that have been modified to use the proposed strategy. These results show that there is a significant improvement in tolerance proportionality in the modified codes at little extra cost over the original versions.     

Space-time discretizing optimal DRP schemes for flow and wave propagation problems

The present paper reports constrained optimization of explicit Runge–Kutta (RK) schemes, coupled with optimal upwind compact scheme to achieve dispersion relation preservation (DRP) property for high performance computing. Essential ideas of optimization employed in arriving at the proposed time integration scheme are extension of the earlier work reported in Rajpoot et al. (J Comput Phys 2010;229:3623–51). This is in turn an application of the correct error evolution equation in Sengupta et al. (J Comput Phys 2007;226:1211–8). Resultant DRP scheme demonstrated the idea for explicit spatial central difference schemes. Present work is similar, extending it for near-spectral accuracy compact schemes. Practical utility of the developed method is demonstrated by solution of model problems and for flow problems by solving Navier–Stokes equation, some of which cannot be solved by conventional schemes, as the problem of rotary oscillation of cylinder.Developed method is calibrated with: (i) flow past a circular cylinder performing rotary oscillation at Re = 150 and (ii) flow inside a 2D lid-driven cavity (LDC) at Reynolds numbers of Re = 1000 and Re = 10,000. Quantitative and qualitative comparisons show excellent match for rotary oscillation cylinder cases with the experimental results of Thiria et al. (J Fluid Mech 2006;560:123–47). Results for LDC for Re = 1000 are compared with that in Botella & Peyret (Comp Fluids 1998;27:421–33) and results for Re = 10,000 are compared with recent published ones showing triangular vortex in the core.     

Iterative schemes for certain time-dependent problems of stochastic optimal control

We prove convergence, with estimates of the rate of convergence, of iterative schemes for discretized versions of systems of parabolic quasi-variational inequalities (QVI). These parabolic QVI arise in problems of stochastic optimal control with discrete control actions (switchings) where the cost functional involves discrete switching costs in addition to continuous rates of cost. We consider two kinds of discretization: discretization with respect to the space variables, and discretization in both time and space.     

Proximal schemes for parabolic optimal control problems with sparsity promoting cost functionals

Fast first-order proximal methods that solve linear and bilinear parabolic optimal control problems with a sparsity cost functional are discussed. Weak convergence of these methods is proved and, for benchmarking purposes, the proposed inexact proximal schemes are compared to an inexact semi-smooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes and to validate the theoretical estimates.     

Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems

The detailed implementation and analysis of a finite element multigrid scheme for the solution of elliptic optimal control problems is presented. A particular focus is in the definition of smoothing strategies for the case of constrained control problems. For this setting, convergence of the multigrid scheme is discussed based on the BPX framework. Results of numerical experiments are reported to illustrate and validate the optimal efficiency and robustness of the performance of the present multigrid strategy.     

P-stable linear symmetric multistep methods for periodic initial-value problems

In this paper we present a new kind of P-stable multistep methods for periodic initial-value problems. From the numerical results obtained by the new method to well-known periodic problems, show the superior efficiency, accuracy, stability of the method presented in this paper.     

The approximate solution of initial-value problems for general Volterra integro-differential equations

We study the application of certain spline collocation methods to Volterra integro-differential equations of orderr where ther-th order derivative of the unknown solution occurs also in the kernel of the integral term. The analysis focuses on the question of the optimal discrete convergence order (at the knots of the approximating spline function).     

Second-order stabilized explicit Runge-Kutta methods for stiff problems

Stabilized Runge-Kutta methods (they have also been called Chebyshev-Runge-Kutta methods) are explicit methods with extended stability domains, usually along the negative real axis. They are easy to use (they do not require algebra routines) and are especially suited for MOL discretizations of two- and three-dimensional parabolic partial differential equations. Previous codes based on stabilized Runge-Kutta algorithms were tested with mildly stiff problems. In this paper we show that they have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value (over 10⁵). We also develop a new procedure to build this kind of algorithms and we derive second-order methods with up to 320 stages and good stability properties. These methods are efficient numerical integrators of very large stiff ordinary differential equations. Numerical experiments support the effectiveness of the new algorithms compared to well-known methods as RKC, ROCK2, DUMKA3 and ROCK4.     

A note on convergence concepts for stiff problems

Most convergence concepts for discretizations of nonlinear stiff initial value problems are based on one-sided Lipschitz continuity. Therefore only those stiff problems that admit moderately sized one-sided Lipschitz constants are covered in a satisfactory way by the respective theory. In the present note we show that the assumption of moderately sized one-sided Lipschitz constants is violated for many stiff problems. We recall some convergence results that are not based on one-sided Lipschitz constants; the concept of singular perturbations is one of the key issues. Numerical experience with stiff problems that are not covered by available convergence results is reported.     

Curvature-based grid step selection for stiff Cauchy problems

A new method of automatic step selection is proposed for the numerical integration of the Cauchy problem for ordinary differential equations. The method is based on using the geometrical characteristics (cuvature and slope) of the integral curve. Formulas have been constructed for the curvature of the integral curve for different choices of multidimensional space. In the two-dimensional case, they turn into well-known formulas, but their general multidimensional form is nontrivial. These formulas have a simple form, are convenient for practical use, and are of independent interest for the differential geometry of multidimensional spaces. For the grids constructed by our method, a procedure of step splitting is proposed that allows one to apply Richardson’s method and to calculate posterior asymptotically precise error estimation for the obtained solution (no such estimates have been found for traditional algorithms of automatic step selection). Therefore, the proposed methods demonstrate significantly superior reliability and validity of the results as compared to calculations by conventional algorithms. In the existing automatic procedures for step selection, steps can be unexpectedly reduced by 2–4 orders of magnitude for no apparent reason. This undermines the reliability of the algorithms. The cause of this phenomenon is explained. The proposed methods are especially effective for highly stiff problems, which is illustrated by examples of calculations.     

High order A-stable numerical methods for stiff problems

Stiff problems pose special computational difficulties because explicit methods cannot solve these problems without severe limitations on the stepsize. This idea is illustrated using a contrived linear test problem and a discretized diffusion problem. Even though the Euler method can solve these problems if the stepsize is small enough, there is no such limitation for the implicit Euler method. To obtain high order A-stable methods, it is traditional to turn to Runge-Kutta methods or to linear multistep methods. Each of these has limitations of one sort or another and we consider, as a middle ground, the use of general linear (or multivalue multistage) methods. Methods possessing the property of inherent Runge-Kutta stability are identified as promising methods within this large class, and an example of one of these methods is discussed. The method in question, even though it has four stages, out-performs the implicit Euler method if sufficient accuracy is required, because of its higher order.     

An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems

An explicit symmetric multistep method is presented in this paper. The new method is exponentially fitted and trigonometrically-fitted and is of algebraic order eight. The effectiveness of the exponential fitting is proved by the application of the new method and the classical one (with constant coefficients) to well-known periodic problems.     

(2,1)-Method for solving stiff nonautonomous problems

An L-stable (2,1)-method with second-order accuracy is developed for solving stiff nonautonomous problems. At each step, the right-hand side of the system, as well as Jacobian matrix decomposition are evaluated only once. Efficiency of the integration method is illustrated using ring modulator calculation as an example.     

Solving inverse initial-value, boundary-value problems via genetic algorithm

There is a growing interest in inverse initial-value, boundary-value (inverse IVBV) problems, and in the development of robust, computationally efficient methods suitable for their solution. Inverse problems are prominent in science and engineering where often an effect is measured and the cause is not known; scientists and engineers observe the response of a system and desire to know the particulars of the system that elicited such a response. IVBV problems result when the equations that govern the behavior of a system are partial differential equations (wave phenomena, diffusion, potential of all kinds, etc.). Thus, inverse IVBV problems stem from systems governed by partial differential equations in which a response has been measured and a characteristic of the system must be computed. In this paper, an approach to solving inverse IVBV problems is presented in which the stated problem is transformed into a nonlinear optimization problem which is then solved using a genetic algorithm. Results are presented demonstrating the effectiveness of this approach for solving inverse problems that result from systems governed by three specific partial differential (1) the heat equation, (2) the wave equation, and (3) Poisson’s equation.     

On the extended one-step schemes for solving stiff systems of ordinary differential equations

Another fourth order extended one-step implicit scheme of solving stiff ordinary differential equations is introduced in this paper, through which, it is shown that such schemes are literally classical implicit Runge-Kutta schemes. Using general theory of Runge-Kutta schemes, stabilities other than A-stability or L-stability are further investigated for the proposed scheme. It is also shown that the parameters involved in such schemes can be better used to reduce the computation cost, making such schemes thus more competitive with traditional ones. Numerical examples are presented showing the competence of such schemes in solving a variety of stiff systems.     

On the iterative inclusion of solutions in initial-value problems for ordinary differential equations

This note contains iterative procedures which bound the solution of initial value problems with the aid of interval analytical methods and possibilities for speeding up convergence.     

Preconditioning in parallel Runge-Kutta methods for stiff initial value problems

From a theoretical point of view, Runge-Kutta methods of collocation type belong to the most attractive step-by-step methods for integrating stiff problems. These methods combine excellent stability features with the property of superconvergence at the step points. Like the initial-value problem itself, they only need the given initial value without requiring additional starting values, and therefore, are a natural discretization of the initial-value problem. On the other hand, from a practical point of view, these methods have the drawback of requiring in each step the solution of a system of equations of dimension sd, s and d being the number of stages and the dimension of the initial-value problem, respectively. In contrast, linear multistep methods, the main competitor of Runge-Kutta methods, require the solution of systems of dimension d. However, parallel computers have changed the scene and have motivated us to design parallel iteration methods for solving the implicit systems in such a way that the resulting methods become efficient step-by-step methods for integrating stiff initial-value problems.     

A new kind of high-efficient and high-accurate P-stable Obrechkoff three-step method for periodic initial-value problems

In order to improve the efficiency and accuracy of the previous Obrechkoff method, in this paper we put forward a new kind of P-stable three-step Obrechkoff method of O(h¹⁰) for periodic initial-value problems. By using a new structure and an embedded high accurate first-order derivative formula, we can avoid time-consuming iterative calculation to obtain the high-order derivatives. By taking advantage of new trigonometrically-fitting scheme we can make both the main structure and the first-order derivative formula to be P-stable. We apply our new method to three periodic problems and compare it with the previous three Obrechkoff methods. Numerical results demonstrate that our new method is superior over the previous ones in accuracy, efficiency and stability.     

The use of quadratures for solving convective and highly stiff transport problems

In this paper we introduce a method for the numerical solutions of initial value problems, that combines finite differences with Simpson’s rule. The effectiveness of the method is proved by solving, in one spatial dimension, a stiff and convection-dominated transport problem. To solve the same problem in two spatial dimensions, the proposed method was used successfully in combination with Strang’s operator decomposition method.