Diagonally implicit Runge-Kutta methods for stiff problems

Diagonally implicit Runge-Kutta methods are examined. It is shown that, for stiff problems, the methods based on the minimization of certain error functions have advantages over other methods; these functions are determined in terms of the errors for simplest model equations. Methods of orders three, four, five, and six are considered.     

Stability of Runge-Kutta methods for quasilinear parabolic problems

We consider a quasilinear parabolic problem

where , , is a family of sectorial operators in a Banach space with fixed domain . This problem is discretized in time by means of a strongly A()-stable, , Runge-Kutta method. We prove that the resulting discretization is stable, under some natural assumptions on the dependence of with respect to . Our results are useful for studying in norms, , many problems arising in applications. Some auxiliary results for time-dependent parabolic problems are also provided.

     

Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems

In a recent series of papers, the class of energy-conserving Runge-Kutta methods named Hamiltonian BVMs (HBVMs) has been defined and studied. Such methods have been further generalized for the efficient solution of general conservative problems, thus providing the class of Line Integral Methods (LIMs). In this paper we derive a further extension, which we name Enhanced Line Integral Methods (ELIMs), more tailored for Hamiltonian problems, allowing for the conservation of multiple invariants of the continuous dynamical system. The analysis of the methods is fully carried out and some numerical tests are reported, in order to confirm the theoretical achievements.     

Explicit adaptive Runge-Kutta methods for stiff and oscillation problems

Explicit Runge-Kutta methods with the coefficients tuned to the problem of interest are examined. The tuning is based on estimates for the dominant eigenvalues of the Jacobian matrix obtained from the results of the preliminary stages. Test examples demonstrate that methods of this type can be efficient in solving stiff and oscillation problems.     

Finite element methods for semilinear parabolic interface problems

In this exposition we study the finite element methods for second-order semilinear parabolic interface problems in two dimensional convex polygonal domains with smooth interface. Both semidiscrete and fully discrete schemes are analyzed. Optimal order error estimates in the L²(0, T; H¹(Ω))-norm are established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

Two-grid finite volume element methods for semilinear parabolic problems

Two-grid finite volume element methods, based on two linear conforming finite element spaces on one coarse grid and one fine grid, are presented and studied for two-dimensional semilinear parabolic problems. With the proposed techniques, solving the nonsymmetric and nonlinear system on the fine space is reduced to solving a symmetric and linear system on the fine space and solving the nonsymmetric and nonlinear system on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. It is proved that the coarse grid can be much coarser than the fine grid. As a result, solving such a large class of semilinear parabolic problems will not be much more difficult than solving one single linearized equation. In the end a numerical example is presented to validate the usefulness and efficiency of the method.     

Two‐grid methods for semilinear interface problems

In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Compact almost automorphic solutions to semilinear Cauchy problems with non-dense domain

In this work we study the existence and uniqueness of compact almost automorphic solutions to a first-order differential equation with a linear part dominated by a Hille-Yosida type operator with non dense domain.     

Towards optimal Runge-Kutta smoothers for multigrid methods for unsteady flow problems

We consider Runge-Kutta smoothers in a dual time stepping multigrid method for unsteady flow problems. These smoothers are easily parallelizable and Jacobian-free, making them very attractive for 3D calculations. Existing methods have been designed for steady flows, leading to slow convergence for unsteady problems. Here we determine the free parameters of the smoother to provide optimal damping for high frequency components for the unsteady linear advection equation. This is compared with an RK smoother designed for steady state problems, as commonly used in CFD codes. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001     

Uniform methods for semilinear problems with attractive boundary turning points

An upwind finite‐difference scheme for the numerical solution of semilinear convection‐diffusion problems with attractive boundary turning points is considered. We show that the maximum nodal error is bounded by a special weighted ℓ₁‐type norm of the truncation error. This result is used to establish uniform convergence with respect to the perturbation parameter on Shishkin meshes.     

Finite element methods for semilinear elliptic problems with smooth interfaces

The purpose of this paper is to study the finite element method for second order semilinear elliptic interface problems in two dimensional convex polygonal domains. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with straight interface triangles [Numer. Math., 79 (1998), pp. 175–202]. For a finite element discretization based on a mesh which involve the approximation of the interface, optimal order error estimates in L ² and H ¹-norms are proved for linear elliptic interface problem under practical regularity assumptions of the true solution. Then an extension to the semilinear problem is also considered and optimal error estimate in H ¹ norm is achieved.     

Runge-Kutta algorithms for oscillatory problems

For the numerical integration of differential equations with oscillatory solutions adapted Runge-Kutta algorithms of up to 4 stages are presented. The coefficients of these methods are chosen such that certain particular oscillatory solutions are computed without truncation errors.

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Zusammenfassung Für die numerische Integration von Differentialgleichungen mit rasch oszillierenden Lösungen werden angepasste Runge-Kutta-Algorithmen mit bis zu 4 Stufen konstruiert. Die Koeffizienten dieser Verfahren werden so gewählt, dass gewisse spezielle oszillierende Lösungen ohne Abbrechfehler berechnet werden.

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This paper is dedicated to the memory of Prof. Dr. E. Stiefel.     

Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations

Runge-Kutta methods are studied when applied to stiff differential equations containing a small stiffness parameter . The coefficients in the expansion of the global error in powers of are the global errors of the Runge-Kutta method applied to a differential algebraic system. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical experiments confirm the results.     

Lie-Butcher theory for Runge-Kutta methods

Runge-Kutta methods are formulated via coordinate independent operations on manifolds. It is shown that there is an intimate connection between Lie series and Lie groups on one hand and Butcher's celebrated theory of order conditions on the other. In Butcher's theory the elementary differentials are represented as trees. In the present formulation they appear as commutators between vector fields. This leads to a theory for the order conditions, which can be developed in a completely coordinate free manner. Although this theory is developed in a language that is not widely used in applied mathematics, it is structurally simple. The recursion for the order conditions rests mainly on three lemmas, each with very short proofs. The techniques used in the analysis are prepared for studying RK-like methods on general Lie groups and homogeneous manifolds, but these themes are not studied in detail within the present paper.     

Exponential decay for semilinear distributed systems with damping

The stability of second order abstract distributed systems with damping and nonlinear perturbations is considered. Sufficient conditions, including unique continuation property assumptions, are formulated to obtain (local, non-uniform and uniform) exponential stability. Applications to the wave and Euler-Bernoulli equations are given.     

Parallelism across the steps in iterated Runge-Kutta methods for stiff initial value problems

For the parallel integration of stiff initial value problems (IVPs) three main approaches can be distinguished: approaches based on parallelism across the problem, on parallelism across the method and on parallelism across the steps. The first type of parallelism does not require special integration methods can be exploited within any available IVP solver. The methodparallel approach received some attention in the case of Runge-Kutta based methods. For these methods, the required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behaviour in the iteration process. Hence, the effective speed-up is rather poor. The step-parallel iteraction process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to solve the four-stage Radau IIA corrector used in our experiments within a few effective iterations per step and to achieve speed-up factors up to 10 with respect to the best sequential codes.The research reported in this paper was partly supported by the Technology Foundation (STW) in the Netherlands.