Multistep Runge-Kutta Research Articles

Development of high-order adaptive multi-step Runge–Kutta–Nyström method for solving special second-order ODEs

Runge–Kutta–Nyström (RKN) methods are extensively used to obtain approximate solutions of ordinary differential equations (ODEs). Specifically, they are widely used to directly solve second-order ODEs of the special form. Although the derivation of new higher-order methods with fewer numbers of function evaluations is of great importance in increasing the precision and effectiveness of the methods, however, this is rarely done due to the difficulty or complexity of some derivations. This study focuses on constructing a 7(5) pair of embedded multi-step Runge–Kutta–Nyström (EMSN) method with lower stages for the numerical solutions of special second-order ODEs. An adaptive step size formulation using an embedded procedure is considered, and the numerical findings reveal that the new embedded pair outperforms existing Runge–Kutta (RK) pairs in terms of the minimum number of functions evaluations.

Multistep Runge–Kutta methods for Volterra integro-differential equations

In this paper, we investigate multistep Runge–Kutta methods for Volterra integro-differential equations. First, we derive order conditions for methods of order p and stage order q=p−1 for p⩽4. Then, stability properties of these methods with respect to the basic and convolution test equations are analyzed. V0-stable methods of order 2 and 3 are obtained. Finally, some numerical experiments are given to verify the obtained theoretical results.

Adaptive multi-step Runge–Kutta–Nyström methods for general second-order ordinary differential equations

In this study, a variable step size formulation of multi-step general Runge–Kutta–Nyström (MSGN) methods to directly integrate general second-order initial value problems (IVPs) is considered. This formula is carried out using an embedded explicit pair where: the higher-order formula is an accurate and the lower-order formula uses to estimate the local error. Numerical results show that the new methods perform much better in terms of functions evaluations, implementation cost and execution time compared to other existing high quality embedded Runge–Kutta (ERK) methods in the literature.

Time Step Restrictions for Strong-Stability-Preserving Multistep Runge–Kutta Discontinuous Galerkin Methods

Time Step Restrictions for Strong-Stability-Preserving Multistep Runge–Kutta Discontinuous Galerkin Methods

Dissipativity of Multistep Runge–Kutta Methods for Nonlinear Neutral Delay-Integro-Differential Equations with Constrained Grid

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear neutral delay-integro-differential equations. We investigate the dissipativity properties of -algebraically stable multistep Runge-Kutta methods with constrained grid. The finite-dimensional and infinite-dimensional dissipativity results of -algebraically stable multistep Runge-Kutta methods are obtained.

Highly stable multistep Runge–Kutta methods for Volterra integral equations

In this paper, we investigate highly stable multistep Runge–Kutta methods for Volterra integral equations. First, the order conditions for order p and stage order $$q=p$$ are presented, and a convergence theorem is given. The numerical stability conditions for the basic and convolution test equations are derived. Then, the methods with one or two stages are studied in detail. Some A-stable and $$V_0$$ -stable m-stage methods with order $$p>m$$ are obtained. For one-stage methods, we also construct $$A_0$$ -stable and -stable methods of orders 3 and 4. Finally, numerical experiments are given to confirm the theoretical results.

On convergence of continuous half-explicit Runge-Kutta methods for a class of delay differential-algebraic equations

In this paper, we propose and investigate continuous Runge-Kutta methods for solving a class of nonlinear differential-algebraic equations (DAEs) with constant delay. Real-life processes that involve simultaneously time-delay effect and constraints are usually described by delay DAEs. Solving delay DAEs is more complicated than solving non-delay ones since we should focus on both the time-delay and DAE aspects. Recently, we have revisited linear multistep methods and Runge-Kutta methods for a class of nonlinear DAEs (without delay) and shown the advantages of appropriately modified discretizations. In this work, we extend the use of half-explicit Runge-Kutta methods to a similar class of structured strangeness-free DAEs with constant delay. Approximation of solutions at delayed time is obtained by continuous extensions of discrete solution, i.e., continuous output formulas. Convergence analysis for continuous Runge-Kutta methods is presented. It is shown that order reduction that may happen with DAEs is avoided if we discretize an appropriately reformulated delay DAE (DDAE) instead of the original one. Difficulties arising in the implementation are discussed as well. Finally, numerical experiments are given for illustration.

An efficient high-order finite difference gas-kinetic scheme for the Euler and Navier–Stokes equations

Based on the temporal evolution of the Navier-Stokes gas distribution function and Weighted Essential Non-Oscillatory (WENO) interpolation, a high-order finite difference gas-kinetic scheme (FDGKS) is constructed. Different from the previous high-order finite volume gas-kinetic method [Li, Xu, and Fu, J. Comput. Phys. vol. 229, pp. 6715 (2010)], which uses a discontinuous initial reconstruction at the cell interface, the present scheme is a finite-difference one with a continuous flow distribution at the grid point.The time-accurate solution of the gas distribution function permits the FDGKS to be a one-step high-order scheme without multi-step Runge-Kutta temporal matching, which significantly reduces the computational time. Many numerical tests in solving one and two-dimensional Euler and Navier-Stokes equations demonstrate that FDGKS is a highly stable, accurate, and efficient scheme, which captures discontinuities without oscillations.

Trigonometrically fitted multi-step RKN methods for second-order oscillatory initial value problems

Trigonometrically fitted multi-step Runge–Kutta–Nyström (TFMSRKN) methods for solving numerically oscillatory special second-order initial value problems are introduced. TFMSRKN methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set {exp(iwt),exp(−iwt)} or equivalently the set {cos (wt), sin (wt)}, where w represents an approximation of the main frequency of the problem. The corresponding order conditions are given and two explicit four-stage TFMSRKN methods with order five are constructed. Stability of the new methods is examined and the regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.

Trigonometrically fitted multi-step Runge-Kutta methods for solving oscillatory initial value problems

In this paper, trigonometrically fitted multi-step Runge-Kutta (TFMSRK) methods for the numerical integration of oscillatory initial value problems are proposed and studied. TFMSRK methods inherit the frame of multi-step Runge-Kutta (MSRK) methods and integrate exactly the problem whose solutions can be expressed as the linear combinations of functions from the set of $\{\exp (\mathrm {i}wt),\exp (-\mathrm {i}wt)\},$ or equivalently the set $\{\cos (wt),\sin (wt)\}$ , where w represents an approximation of the main frequency of the problem. The general order conditions are given and four new explicit TFMSRK methods with order three and four, respectively, are constructed. Stability of the new methods is examined and the corresponding regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.

Multi-step Runge–Kutta–Nyström methods for special second-order initial value problems

In this paper, multi-step Runge–Kutta–Nyström methods for the numerical integration of special second-order initial value problems are proposed and studied. These methods include classical Runge–Kutta–Nyström methods as special cases. General order conditions are derived by using the theory of B-series based on the set of special Nyström-trees, and two explicit methods with order five and six, respectively, are constructed. Numerical results show that our new methods are more efficient in comparison with classical Runge–Kutta–Nyström methods and other well-known high quality methods proposed in the scientific literature.

Optimised prefactored compact schemes for linear wave propagation phenomena

A family of space- and time-optimised prefactored compact schemes are developed that minimise the computational cost for given levels of numerical error in wave propagation phenomena, with special reference to aerodynamic sound. This work extends the approach of Pirozzoli [1] to the MacCormack type prefactored compact high-order schemes developed by Hixon [2], in which their shorter Padé stencil from the prefactorisation leads to a simpler enforcement of numerical boundary conditions. An explicit low-storage multi-step Runge–Kutta integration advances the states in time. Theoretical predictions for spatial and temporal error bounds are derived for the cost-optimised schemes and compared against benchmark schemes of current use in computational aeroacoustic applications in terms of computational cost for a given relative numerical error value. One- and two-dimensional test cases are presented to examine the effectiveness of the cost-optimised schemes for practical flow computations. An effectiveness up to about 50% higher than the standard schemes is verified for the linear one-dimensional advection solver, which is a popular baseline solver kernel for computational physics problems. A substantial error reduction for a given cost is also obtained in the more complex case of a two-dimensional acoustic pulse propagation, provided the optimised schemes are made to operate close to their nominal design points.

Explicit strong stability preserving multistep Runge–Kutta methods

High-order spatial discretizations with strong stability properties (such as monotonicity) are desirable for the solution of hyperbolic PDEs. Methods may be compared in terms of the strong stability preserving (SSP) time-step. We prove an upper bound on the SSP coefficient of explicit multistep Runge–Kutta methods of order two and above. Order conditions and monotonicity conditions for such methods are worked out in terms of the method coefficients. Numerical optimization is used to find optimized explicit methods of up to five steps, eight stages, and tenth order. These methods are tested on the advection and Buckley-Leverett equations, and the results for the observed total variation diminishing and positivity preserving time-step are presented.

Numerical dissipativity of neutral integro-differential equations with delay

ABSTRACTMany dynamical systems in physics and engineering are characterized by the property of possessing a bounded absorbing set which all trajectories enter in a finite time and thereafter remain inside. It is highly important to analyse whether or not the numerical methods for solving these dynamical systems inherit such property of the underlying systems. In this paper, it is proved that, under some assumptions, a multistep Runge–Kutta (abbrev. MRK) method is dissipative when it is applied to a class of neutral integro-differential equations with delay, provided it is -algebraically stable. Numerical examples are given to confirm our theoretical results.

A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry

We develop a high-order kinetic scheme for entropy-based moment models of a one-dimensional linear kinetic equation in slab geometry. High-order spatial reconstructions are achieved using the weighted essentially non-oscillatory (WENO) method, and for time integration we use multi-step Runge-Kutta methods which are strong stability preserving and whose stages and steps can be written as convex combinations of forward Euler steps. We show that the moment vectors stay in the realizable set using these time integrators along with a maximum principle-based kinetic-level limiter, which simultaneously dampens spurious oscillations in the numerical solutions. We present numerical results both on a manufactured solution, where we perform convergence tests showing our scheme converges of the expected order up to the numerical noise from the numerical optimization, as well as on two standard benchmark problems, where we show some of the advantages of high-order solutions and the role of the key parameter in the limiter.

Dissipativity of multistep runge-kutta methods for nonlinear volterra delay-integro-differential equations

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. We investigate the dissipativity properties of (k, l)-algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid. The finitedimensional and infinite-dimensional dissipativity results of (k, l)-algebraically stable Runge-Kutta methods are obtained.

Numerical Simulation of Transverse Injection of Circular Jets into Turbulent Supersonic Streams

The flowfields arising from transverse injection of sonic air jets through circular holes on flat plates into supersonic streams have been computed. Favre-averaged Navier–Stokes equations for compressible flow coupled with k–! equations were solved with a finite-volume method. The approximate Riemann solver due to Roe has been used for obtaining inviscid fluxes at cell interfaces, with monotone upwind schemes for scalar conservation laws preprocessing for high resolution and a multistep Runge–Kutta method for time integration to steady states. A detailed comparison has been made with recent laser Doppler velocimetry measurements of velocity and turbulence kinetic energy on the symmetry plane and two crossflow planes. Wall pressures were compared with those obtained with pressure-sensitive paint from the same experimental facility. Wall pressures and features at higher crossflow Mach numbers were examined by simulating a second set of experiments. Computations capture various flow features and show very good qualitative agreement with both sets of experiments. Quantitative agreement is very good in the upstream regions and close to the jet. Small differences appear in the downstream portion as the jet develops. The changes in flow structure with increasing injection pressure suggested by sketches of oil flows are also obtained in the simulations

Equilibrium attractive properties of a class of multistep Runge–Kutta methods

The main purpose of this paper is to discuss the equilibrium attractive properties of a class of multistep Runge–Kutta methods for initial value problems of ordinary differential equations. Some algebraic conditions insuring the equilibrium attractivity are given, and some methods satisfying these algebraic conditions are constructed. Some numerical examples confirm our results.

Dissipativity of multistep runge-kutta methods for dynamical systems with delays

This paper is concerned with the numerical solution of dissipative initial value problems with delays by multistep Runge-Kutta methods. We investigate the dissipativity properties of ( k, l)-algebraically stable multistep Runge-Kutta methods with constrained grid and linear interpolation procedure. In particular, it is proved that an algebraically stable, irreducible multistep Runge-Kutta method is dissipative for finite-dimensional dynamical systems with delays, which extends and unifies some extant results. In addition, we obtain dissipativity results of A-stable linear multistep methods by using the relationship between one-leg methods and linear multistep methods.

Stability of multistep Runge-Kutta methods for systems of functional-differential and functional equations

This paper is concerned with the stability of multistep Runge-Kutta methods applied to linear systems of functional-differential and functional equations. The adaptation of multistep Runge-Kutta methods is considered. It is proved that, under some reasonable assumptions, A-stability multistep Runge-Kutta methods can inherit the asymptotic stability of underlying linear systems.