Explicit One-step Methods Research Articles

A general class of second-order [formula omitted]-stable explicit numerical methods for stiff problems

In this paper, we propose a simple approach to the construction of a general class of L-stable explicit second-order one-step methods for solving stiff problems. These methods are nonlinear and derive from a novel approximation for the right-hand side functions of differential equations inspired by the nonstandard finite difference methodology introduced by Mickens. Through rigorous mathematical analysis, it is proved that the proposed numerical methods are not only explicit and L-stable, but also convergent of order two. Therefore, they are suitable and efficient to solve stiff problems.The proposed numerical methods generalize and improve a nonstandard explicit integration scheme for initial value problems formulated by Ramos (2007). Moreover, the present approach can be extended to construct A-stable and L-stable high-order explicit one-step methods for differential equations.Finally, the theoretical findings and advantages of the developed numerical methods are supported and illustrated by a series of numerical experiments in which stiff problems are considered.

ON THE CONSISTENCY AND CONVERGENCE OF CLASSICAL RICHARDSON EXTRAPOLATION AS APPLIED TO EXPLICIT ONE-STEP METHODS

The consistency of the classical Richardson extrapolation (CRE), a simple and robust computational device, is analysed for the case where the underlying method is an explicit one-step numerical method for ordinary differential equations with order of consistency one or two. It is shown in the classical framework that the CRE increases the order of consistency by one. The convergence of the method is proved by the assumption that the time-stepping operator of the base method has the Lipschitz property in its second argument.

Numerical methods for the hyperbolic Monge-Ampère equation based on the method of characteristics

We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation (PDE) in two independent variables. The MOC gives rise to two mutually coupled systems of ordinary differential equations (ODEs). As a special case we consider the Monge–Ampère (MA) equation, for which we present a general method of determining the location and number of required boundary conditions. We solve the systems of ODEs using explicit one-step methods (Euler, Runge-Kutta) and spline interpolation. Reformulation of the Monge–Ampère equation as an integral equation yields via its residual a proxy for the error of the numerical solution. Numerical examples demonstrate the performance and convergence of the methods.

Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. By selecting the rank of the tensor manifold adaptively to satisfy stability and accuracy requirements, we prove convergence of a wide range of step-truncation methods, including explicit one-step and multi-step methods. These methods are very easy to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which, have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a Fokker-Planck equation with spatially dependent drift on a flat torus of dimension two and four.

Computation of Optimal Linear Strong Stability Preserving Methods Via Adaptive Spectral Transformations of Poisson–Charlier Measures

Strong stability preserving (SSP) coefficients govern the maximally allowable step-size at which positivity or contractivity preservation of integration methods for initial value problems is guaranteed. In this paper, we show that the task of computing optimal linear SSP coefficients of explicit one-step methods is, to a certain extent, equivalent to the problem of characterizing positive quadratures with integer nodes with respect to Poisson–Charlier measures. Using this equivalence, we provide sharp upper and lower bounds for the optimal linear SSP coefficients in terms of the zeros of generalized Laguerre orthogonal polynomials. This in particular provides us with a sharp upper bound for the optimal SSP coefficients of explicit Runge–Kutta methods. Also based on this equivalence, we propose a highly efficient and stable algorithm for computing these coefficients, and their associated optimal linear SSP methods, based on adaptive spectral transformations of Poisson–Charlier measures. The algorithm possesses the remarkable property that its complexity depends only on the order of the method and thus is independent of the number of stages. Our results are achieved by adapting and extending an ingenious technique by Bernstein (Acta Math 52:1–66, 1928) in his seminal work on absolutely monotonic functions. Moreover, the techniques introduced in this work can be adapted to solve the integer quadrature problem for any positive discrete multi-parametric measure supported on \({\mathbb {N}}\) under some mild conditions on the zeros of the associated orthogonal polynomials.

Construction and Analysis of Explicit Adaptive One-Step Methods for Solving Stiff Problems

The paper considers the construction of adaptive methods based on the explicit Runge–Kutta stages. The coefficients of these methods are adjusted to the problem being solved, using component-wise estimates of the eigenvalues of the Jacobi matrix with the maximum absolute values. Such estimates can be easily obtained at the stages of the explicit method, which practically does not require additional calculations. The effect of computational errors and stiffness of the problem on the stability and accuracy of the numerical solution is studied. The analysis allows one to construct efficient explicit methods that are not inferior to implicit methods in solving many stiff problems. New nested pairs of adaptive methods are proposed, and the results of numerical experiments are presented.

Improving locality of explicit one-step methods on GPUs by tiling across stages and time steps

The performance of explicit parallel methods solving large systems of ordinary differential equations (ODEs) on GPUs is often memory bound. Therefore, locality optimizations, such as kernel fusion, are desirable. This paper exploits a special property of a large class of right-hand-side (RHS) functions to enable the fusion of computations of blocks of components of dependent stages of the method. This allows the derivation of tilings of the stages not only within one time step, but also spanning several successive time steps. Our approach is based on a representation of the ODE method by a data flow graph and allows efficient GPU code with fused kernels to be generated automatically for user-defined tilings. In particular, we investigate two generalized tiling strategies, trapezoidal and hexagonal tiling, and two different partitionings, which are evaluated experimentally for several different high- and low-order Runge–Kutta (RK) methods.

A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations

A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [0,a] with a singularity of the first kind in the point x=0 is proposed. Using the substitution of the independent variable x=et, we reduce the original initial value problem to the one on the interval (−∞,ln⁡a]. On some finite irregular grid {tn∈(−∞,ln⁡a], n=0,1,...,N,tN=ln⁡a} Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t0, new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown.

An explicit one-step multischeme sixth order method for systems of special structure

Structure based partitioning of a system of ordinary differential equations is considered. A general form of the explicit multischeme Runge–Kutta type method for such systems is presented. Order conditions and simplifying conditions are written down. An algorithm of derivation of the sixth order method with seven stages and reuse with two free parameters is given. It embeds a fourth order error estimator. Numerical comparison to the Dormand–Prince method with the same computation cost but of lower order is performed.

On Numerical Modeling of the Multidimensional Dynamic Systems under Random Perturbations with the 1.5 and 2.0 Orders of Strong Convergence

The article is devoted to explicit one-step numerical methods with strong orders 1.0, 1.5, and 2.0 of convergence for Ito stochastic differential equations with multidimensional and non-commutative noise. For numerical modeling of iterated Ito stochastic integrals with multiplicities 1 to 4 we use the method of multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k=1,2,3,4.$ The article is addressed to engineers who use numerical modeling in stochastic control and for solving the nonlinear filtering problem.

Numerical analysis for some autonomous stochastic delay differential equations

Numerical solution of stochastic delay differential equations is studied by using explicit one-step methods. One-step method is asymptotically zero-stable in the quadratic mean square sense.

Efficient direct multiple shooting for nonlinear model predictive control on long horizons

We address direct multiple shooting based algorithms for nonlinear model predictive control, with a focus on problems with long prediction horizons. We describe different efficient multiple shooting variants with a computational effort that is only linear in the horizon length. Proposed techniques comprise structure exploiting linear algebra on the one hand, and approximation of derivative information in an adjoint Sequential Quadratic Programming method on the other hand. For explicit one-step methods for ordinary differential equations we address the issue of consistent and fast generation of both forward and adjoint derivatives of dynamic process models according to the principle of Internal Numerical Differentiation. We discuss the applicability of the proposed methods at the example of three benchmark problems. These have recently been addressed in literature and serve to evaluate the relative performance of each of the proposed methods for both off-line optimal control and on-line nonlinear model predictive control. Throughout, we compare against results published for a recently proposed collocation approach based on finite elements.

Automatic step size and order control in explicit one-step extrapolation methods

A theory is presented for implicit one-step extrapolation methods for ordinary differential equations. The computational schemes used in such methods are based on the implicit Runge-Kutta methods. An efficient implementation of implicit extrapolation is based on the combined step size and order control. The emphasis is placed on calculating and controlling the global error of the numerical solution. The aim is to achieve the user-prescribed accuracy in an automatic mode (ignoring round-off errors). All the theoretical conclusions of this paper are supported by the numerical results obtained for test problems.

A higher order local linearization method for solving ordinary differential equations

The local linearization (LL) method for the integration of ordinary differential equations is an explicit one-step method that has a number of suitable dynamical properties. However, a major drawback of the LL integrator is that its order of convergence is only two. The present paper overcomes this limitation by introducing a new class of numerical integrators, called the LLT method, that is based on the addition of a correction term to the LL approximation. In this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the LL method. In particular, it is proved that the LLT method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence. The performance of the introduced method is further illustrated through computer simulations.

A class of explicit one-step methods of order two for stiff problems

In this paper we introduce a new class of explicit one-step methods of order 2 that can be used for solving stiff problems. This class constitutes a generalization of the two-stage explicit Runge– Kutta methods, with the property of having an A-stability region that varies during the integration in accordance with the accuracy requirements. Some numerical experiments on classical stiff problems are presented.

A class of explicit one-step methods of order two for stiff problems

In this paper we introduce a new class of explicit one-step methods of order 2 that can be used for solving stiff problems. This class constitutes a generalization of the two-stage explicit Runge– Kutta methods, with the property of having an A-stability region that varies during the integration in accordance with the accuracy requirements. Some numerical experiments on classical stiff problems are presented.

An Explicit One-Step Method for Stiff Problems

In this paper we introduce an explicit one-step method that can be used for solving stiff problems. This method can be viewed as a modification of the explicit Euler method that allows to reduce the stiffness in some sense. Some numerical experiments on linear stiff problems and on the Van der Pol's equation show the effectiveness of the method.

Two low accuracy methods for stiff systems

Two low accuracy explicit one-step methods for stiff ordinary differential equations are extended directly to solve systems of equations. Some defects of the component form of these methods are avoided. To perform these, a new set of vector computations are introduced. Some numerical experiments are presented to show the superiority of the new methods.

The vector form of a sixth-order A-stable explicit one-step method for stiff problems

A sixth-order A-stable explicit one-step method for stiff ordinary differential equations is extended directly to solve systems of equations. Some defects of the component form of this method are avoided. To perform these, a new set of vector computations are introduced. Some numerical experiments are presented to show its superiority.

Numerical Analysis of Explicit One-Step Methods for Stochastic Delay Differential Equations

AbstractWe consider the problem of strong approximations of the solution of stochastic differential equations of Itô form with a constant lag in the argument. We indicate the nature of the equations of interest, and give a convergence proof in full detail for explicit one-step methods. We provide some illustrative numerical examples, using the Euler–Maruyama scheme.