Continuous block implicit hybrid one-step methods for ordinary and delay differential equations

A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and Aω-stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.     

Adjoint methods for static Hamilton�CJacobi equations

We use the adjoint methods to study the static Hamilton?CJacobi equations and to prove the speed of convergence for those equations. The main new ideas are to introduce adjoint equations corresponding to the formal linearizations of regularized equations of vanishing viscosity type, and from the solutions ?? ^?? of those we can get the properties of the solutions u of the Hamilton?CJacobi equations. We classify the static equations into two types and present two new ways to deal with each type. The methods can be applied to various static problems and point out the new ways to look at those PDE.     

Boundedness of solutions of difference equations and application to numerical solution of Volterra integral equations of the second kind

We study the difference equations obtained when some numerical methods for Volterra integral equations of the second kind are applied to the linear test problem y(t) = 1 + ∝₀^t (λ + μt + vs) y(s) ds, t ⩾ 0, with fixed stepsize h. The resulting difference equations are of Poincaré type and we formulate a criterion for boundedness of solutions of these equations if the associated characteristic polynomial is a simple von Neumann polynomial. This result is then used in stability analysis of reducible quadrature methods for Volterra integral equations.     

Variational iteration technique for solving nonlinear equations

In this paper, we use the variational iteration technique to suggest some new iterative methods for solving nonlinear equations f(x)=0. We also discuss the convergence criteria of these new iterative methods. Comparison with other similar methods is also given. These new methods can be considered as an alternative to the Newton method. We also give several examples to illustrate the efficiency of these methods. This technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations.     

Natural Volterra Runge-Kutta methods

A very general class of Runge-Kutta methods for Volterra integral equations of the second kind is analyzed. Order and stage order conditions are derived for methods of order p and stage order q = p up to the order four. We also investigate stability properties of these methods with respect to the basic and the convolution test equations. The systematic search for A- and V ₀-stable methods is described and examples of highly stable methods are presented up to the order p = 4 and stage order q = 4.     

Linear functional equations of the first, second, and third kind in L 2

Under consideration are the functional equations of the first, second, and third kind with operators in wide classes of linear continuous operators in L ₂ containing all integral operators. We propose methods for reducing these equations by linear invertible changes either to linear integral equations of the first kind with nuclear operators or to equivalent linear integral equations of the second kind with quasidegenerate Carleman kernels. Some various approximate methods of solution are applicable to the so-obtained integral equations.     

Collocation methods for second-order volterra integro-differential equations

We study the numerical solution of second-order Volterra integro-differential equations by means of collocation techniques in certain polynomial spline spaces. Suitable discretization of the resulting collocation equation yields implicit methods which may be viewed as extensions of m-stage implicit Runge-Kutta-Nyström methods for initial-value problems of second-order ordinary differential equations to second-order integro-differential equations. The attainable order of (local) convergence of these methods is analyzed in detail.     

Hybrid solvers for composition and splitting methods

Composition and splitting are useful techniques for constructing special purpose integration methods for numerically solving many types of differential equations. In this article we will review these methods and summarise the essential ingredients of an implementation that has recently been added to a framework for solving differential equations in Mathematica.     

Implicit peer methods for large stiff ODE systems

Implicit two-step peer methods are introduced for the solution of large stiff systems. Although these methods compute s-stage approximations in each time step one-by-one like diagonally-implicit Runge-Kutta methods the order of all stages is the same due to the two-step structure. The nonlinear stage equations are solved by an inexact Newton method using the Krylov solver FOM (Arnoldi??s method). The methods are zero-stable for arbitrary step size sequences. We construct different methods having order p=s in the multi-implicit case and order p=s?1 in the singly-implicit case with arbitrary step sizes and s??5. Numerical tests in Matlab for several semi-discretized partial differential equations show the efficiency of the methods compared to other Krylov codes.     

A fractional spectral method with applications to some singular problems

In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Müntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x ^1/λ ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods.     

Variable multistep methods for delay differential equations

In this paper, variable stepsize multistep methods for delay differential equations of the type y(t) = f(t, y(t), y(t − τ)) are proposed. Error bounds for the global discretization error of variable stepsize multistep methods for delay differential equations are explicitly computed. It is proved that a variable multistep method which is a perturbation of strongly stable fixed step size method is convergent.     

Strongly elliptic pseudodifferential equations on the sphere with radial basis functions

Spherical radial basis functions are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the Galerkin and collocation methods. A salient feature of the paper is a unified theory for error analysis of both approximation methods.     

On the efficiency of a class of a-stable methods

The numerical solution of systems of differential equations of the formB dx/dt=σ(t)Ax(t)+f(t),x(0) given, whereB andA (withB and —(A+A ^T) positive definite) are supposed to be large sparse matrices, is considered.A-stable methods like the Implicit Runge-Kutta methods based on Radau quadrature are combined with iterative methods for the solution of the algebraic systems of equations.     

Coupled nonlinear stochastic differential equations

Systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co-workers. Examples include two coupled Riccati equations, coupled linear equations, stochastic coupled equations with product terms, and n coupled stochastic differential equations.     

A class of Steffensen type methods with optimal order of convergence

In this paper, a family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided differences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index equal to 1.587. Kung and Traub conjectured that the order of convergence of any multipoint method without memory cannot exceed the bound 2^d-1, where d is the number of functional evaluations per step. The new class of methods agrees with this conjecture for the case d=3. Numerical examples are made to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other ones.     

Convergence and optimization of the parallel method of simultaneous directions for the solution of elliptic problems

For the solution of elliptic problems, fractional step methods and in particular alternating directions (ADI) methods are iterative methods where fractional steps are sequential. Therefore, they only accept parallelization at low level. In [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708], Lu et al. proposed a method where the fractional steps can be performed in parallel. We can thus speak of parallel fractional step (PFS) methods and, in particular, simultaneous directions (SDI) methods. In this paper, we perform a detailed analysis of the convergence and optimization of PFS and SDI methods, complementing what was done in [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708]. We describe the behavior of the method and we specify the good choice of the parameters. We also study the efficiency of the parallelization. Some 2D, 3D and high-dimensional tests confirm our results.     

Derivation of projection methods from formal integration of the Navier-Stokes equations

Projection methods constitute a class of numerical methods for solving the incompressible Navier-Stokes equations. These methods operate using a two-step procedure in which the zero-divergence constraint on the velocity is first relaxed while the velocity evolves, then after a certain period of time the resulting velocity field is projected onto a divergence-free subspace. Although these methods can be quite efficient, there have been certain concerns regarding their formulation. In this paper we show how a formal integration of the Navier-Stokes equations leads to a new and general procedure for the derivation of projection methods. By following this procedure, we show how each of three practical projection methods approximates a system of equations that is equivalent to the Navier-Stokes equations. We also show how the auxiliary boundary conditions required in projection methods are related to the physical boundary conditions. These results should allay the concerns regarding the legitimacy of projection methods, and may assist in their future development.     

Simultaneous solutions to diagonal equations over finite fields

An elementary proof of the Weil conjectures is given for the special case of a non-singular pair of diagonal equations over a finite field. The number of simultaneous solutions to an arbitrary number of diagonal equations over GF(q) is also estimated by the same classical methods.     

A note on the efficient implementation of implicit methods for ODEs

The use of implicit methods for ODEs, e.g. implicit Runge-Kutta schemes, requires the solution of nonlinear systems of algebraic equations of dimension s · m, where m is the size of the continuous differential problem to be approximated. Usually, the solution of this system represents the most time-consuming section in the implementation of such methods. Consequently, the efficient solution of this section would improve their performance. In this paper, we propose a new iterative procedure to solve such equations on sequential computers.     

A nonlinear optimization approach to the construction of general linear methods of high order

We describe the construction of diagonally implicit multistage integration methods of order and stage order p = q = 7 and p = q = 8 for ordinary differential equations. These methods were obtained using state-of-the-art optimization methods, particularly variable-model trust-region least-squares algorithms.