Starting algorithms for low stage order RKN methods

When second order differential equations are solved with Runge-Kutta-Nyström methods, the computational effort is dominated by the cost of solving the nonlinear system. That is why it is important to have good starting values to begin the iterations. In this paper we consider a type of starting algorithms without additional computational cost. We study the general order conditions and the maximum order achieved when the Runge-Kutta-Nyström method satisfies some simplifying assumptions.     

SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Traditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge‐Kutta methods (also called Runge–Kutta–Chebyshev methods) were proposed to try to avoid these difficulties. The Runge–Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In Martín‐Vaquero and Janssen [Comput Phys Commun 180 (2009), 1802–1810], we showed that previous codes based on stabilized Runge–Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth‐order polynomials. Here, we develop a new method based on second‐order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well‐known second‐order methods such as RKC and ROCK2. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Continuous extensions to Nyström methods for second order initial value problems

In this work we consider interpolants for Nyström methods, i.e., methods for solving second order initial value problems. We give a short introduction to the theory behind the discrete methods, and extend some of the work to continuous, explicit Nyström methods. Interpolants for continuous, explicit Runge-Kutta methods have been intensively studied by several authors, but there has not been much effort devoted to continuous Nyström methods. We therefore extend some of the work by Owren.     

High‐resolution relaxation scheme for the two‐dimensional Riemann problems in gas dynamics

We present a new relaxation method for the numerical approximation of the two‐dimensional Riemann problems in gas dynamics. The novel feature of the technique proposed here is that it does not require either a Riemann solver or a characteristics decomposition. The high resolution of the method is achieved by using a third‐order reconstruction for the space discretization and a third‐order TVD Runge‐Kutta scheme for the time integration. Numerical experiments, using several configurations of Riemann problems in gas dynamics, are included to confirm the high resolution of the new relaxation scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

ERKN methods for long-term integration of multidimensional orbital problems

This paper is devoted to introducing ERKN methods for long-term integration of multidimensional orbital problems. For the general multidimensional perturbed oscillators y^″+My=f(t,y)$y^{″}+\mathit{My}=f(t\text{,}y)$ with M∈R^m×m$M\in {\mathbb{R}}^{m\times m}$, the extended Runge–Kutta–Nyström (ERKN) methods are proposed by Wu et al. [X. Wu, X. You, W. Shi, B. Wang, ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Commun. 181 (2010) 1873–1887]. These methods exactly integrate the multidimensional unperturbed oscillators and are highly efficient when the perturbing forces are small. In this paper, we pay attention to the applications of ERKN methods to multidimensional orbital problems. Numerical experiments accompanied demonstrate that for long-term integration of multidimensional orbital problems the multidimensional ERKN methods are more efficient compared with high-quality codes proposed in the scientific literature. In particular, when an orbital problem under consideration is a Hamiltonian system, the symplectic ERKN methods preserve the Hamiltonian very well, and has better accuracy than the high-quality codes with the same computational cost.     

New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients

Two‐derivative Runge‐Kutta methods are Runge‐Kutta methods for problems of the form y′ = f(y) that include the second derivative y′′ = g(y) = f ′(y)f(y) and were developed in the work of Chan and Tsai. In this work, we consider explicit methods and construct a family of fifth‐order methods with three stages of the general case that use several evaluations of f and g per step. For problems with oscillatory solution and in the case that a good estimate of the dominant frequency is known, methods with frequency‐dependent coefficients are used; there are several procedures for constructing such methods. We give the general framework for the construction of methods with variable coefficients following the approach of Simos. We modify the above family to derive methods with frequency‐dependent coefficients following this approach as well as the approach given by Vanden Berghe. We provide numerical results to demonstrate the efficiency of the new methods using three test problems.     

A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains

The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included.     

Numerical treatment of sth order boundary value problems by solving second kind Fredholm integral equations

A Nyström method is proposed for solving Fredholm integral equations equivalent to boundary value problems of order s with complete differential equations. The stability and the convergence of the proposed procedure are proved. Some numerical examples are provided in order to illustrate the accuracy of the method and to compare the procedure with some other ones given in the literature.     

High‐order Padé and singly diagonally Runge‐Kutta schemes for linear ODEs,application to wave propagation problems

In this article, we address the problem of constructing high‐order implicit time schemes for wave equations. We consider two classes of one‐step A‐stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based on the diagonal Padé approximant of exponential function. For this class, the obtained schemes have the same stability function as Gauss Runge‐Kutta (Gauss RK) schemes. They have the advantage to involve the solution of smaller linear systems at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge‐Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second‐class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy for both classes of schemes, and the great interest of considering high‐order time schemes that are faster. The diagonal Padé schemes seem to be more accurate and more robust.     

Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator

The resonator problem for a positive branch confocal unstable resonator reduces to a Fredholm homogeneous integral equation of the second kind, whose numerical solution here is based on a sequence of algebraic eigenvalue problems. We compare two algorithms for the solution of an optical resonator problem. These are obtained by (i) successive degenerate kernel approximation by Taylor polynomials of the Fredholm kernel and (ii) Nyström’s method with Simpson’s rule as the subordinate numerical integration method. The numerical results arising from these routines compare well with other published results, and have the added advantage of simplicity and easy adaptability to other resonator problems.     

Lattice-Nyström method for Fredholm integral equations of the second kind with convolution type kernels

We consider Fredholm integral equations of the second kind of the form , where g and k are given functions from weighted Korobov spaces. These spaces are characterized by a smoothness parameter α>1 and weights γ₁≥γ₂≥. The weight γ_j moderates the behavior of the functions with respect to the jth variable. We approximate f  by the Nyström method using n rank-1 lattice points. The combination of convolution and lattice group structure means that the resulting linear system can be solved in O(nlogn) operations. We analyze the worst case error measured in sup norm for functions g in the unit ball and a class of functions k in weighted Korobov spaces. We show that the generating vector of the lattice rule can be constructed component-by-component to achieve the optimal rate of convergence O(n^-α/2+δ), δ>0, with the implied constant independent of the dimension d under an appropriate condition on the weights. This construction makes use of an error criterion similar to the worst case integration error in weighted Korobov spaces, and the computational cost is only O(nlognd) operations. We also study the notion of QMC-Nyström tractability: tractability means that the smallest n needed to reduce the worst case error (or normalized error) to is bounded polynomially in ^-1 and d; strong tractability means that the bound is independent of d. We prove that strong QMC-Nyström tractability in the absolute sense holds iff , and QMC-Nyström tractability holds in the absolute sense iff .     

Optimized higher order time discretization of second order hyperbolic problems: Construction and numerical study

We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail. The optimal schemes are validated through various numerical results.     

High‐order finite difference schemes for numerical solutions of the generalized Burgers–Huxley equation

In this article, up to tenth‐order finite difference schemes are proposed to solve the generalized Burgers–Huxley equation. The schemes based on high‐order differences are presented using Taylor series expansion. To establish the numerical solutions of the corresponding equation, the high‐order schemes in space and a fourth‐order Runge‐Kutta scheme in time have been combined. Numerical experiments have been conducted to demonstrate the high‐order accuracy of the current algorithms with relatively minimal computational effort. The results showed that use of the present approaches in the simulation is very applicable for the solution of the generalized Burgers–Huxley equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithms are seen to be very good alternatives to existing approaches for such physical applications. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1313‐1326, 2011     

Solution operators of three time variables for fractional linear problems

It is well known that the time fractional equation where is the fractional time derivative in the sense of Caputo of u does not generate a dynamical system in the standard sense. In this paper, we study the algebraic properties of the solution operator T(t,s,τ) for that equation with u(s) = v. We apply this theory to linear time fractional PDEs with constant coefficients. These equations are solved by the Fourier multiplier techniques. It appears that their solution exhibits some singularity, which leads us to introduce a new kind of solution for abstract time fractional problems. Copyright © 2016 John Wiley & Sons, Ltd.     

A new class of second order linearly implicit fractional step methods

A new family of linearly implicit fractional step methods is proposed and analysed in this paper. The combination of one of these time integrators with a suitable spatial discretization permits a very efficient numerical solution of semilinear parabolic problems. The main quality of this new family of methods, compared to other existing time integrators of this type, is that they are stable when the spatial differential operator is decomposed in a number m$m$ of “simpler” operators which do not necessarily commute. We prove that these methods satisfy this general stability result as well as they are second order consistent. Both consistency and stability are proven for an operator splitting in an arbitrary number m$m$ of terms (m?2$m?2$). Finally, a numerical experiment illustrates these theoretical results in the last section of the paper.     

Well-posedness of second order evolution equation on discrete time

We characterize the well-posedness for second order discrete evolution equations in unconditional martingale difference spaces by means of Fourier multipliers and R-boundedness properties of the resolvent operator which defines the equation. Applications to semilinear problems are given.     

Nyström discretization of parabolic boundary integral equations

A Nyström method for the discretization of thermal layer potentials is proposed and analyzed. The method is based on considering the potentials as generalized Abel integral operators in time, where the kernel is a time dependent surface integral operator. The time discretization is the trapezoidal rule with a corrected weight at the endpoint to compensate for singularities of the integrand. The spatial discretization is a standard quadrature rule for surface integrals of smooth functions. We will discuss stability and convergence results of this discretization scheme for second-kind boundary integral equations of the heat equation. The method is explicit, does not require the computation of influence coefficients, and can be combined easily with recently developed fast heat solvers.