General linear methods – p. 1/58 General linear methods The name “general linear methods ” applies to a large family of numerical methods for ordinary differential equations. General linear methods – p. 2/58 General linear methods The name “general linear methods ” applies to a large family of numerical methods for ordinary differential equations. Runge-Kutta methods are examples of these methods. General linear methods – p. 2/58 General linear methods The name “general linear methods ” applies to a large family of numerical methods for ordinary differential

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Discretization methods for ordinary differential equations are usually not exact; they commit an error at every step of the algorithm. All these errors combine to form the global error, which is the error in the final result. The global error is the subject of this thesis. In the first half of the thesis, accurate a priori estimates of the global error are derived. Three different approaches are followed: to combine the effects of the errors committed at every step, to expand the global error in an asymptotic series in the step size, and to use the theory of modified equations. The last approach, which is often the most useful one, yields an estimate which is correct up to a term of order h 2p, where h denotes the step size and p the order of the numerical method. This result is then applied to estimate the global error for the Airy equa-tion (and related oscillators that obey the Liouville–Green approximation) and the Emden–Fowler equation. The latter example has the interesting feature that it is not sufficient to consider only the leading global error term, because subsequent terms of higher order in the step size may grow faster in time. The second half of the thesis concentrates on minimizing the global error by varying the step size. It is argued that the correct objective function is the norm of the global error over the entire integration interval. Specifically, the L2 norm and the L ∞ norm are studied. In the former case, Pontryagin’s Minimum Principle converts the problem to a boundary value problem, which may be solved analyti-cally or numerically. When the L ∞ norm is used, a boundary value problem with a complementarity condition results. Alternatively, the Exterior Penalty Method may be employed to get a boundary value problem without complementarity condition, which can be solved by standard numerical software. The theory is illustrated by calculating the optimal step size for solving the Dahlquist test equation and the Kepler problem. i

The efficiency of numerical time–stepping methods for dynamical systems is greatly enhanced by automatic time step variation. In this paper we present and discuss three different approaches to step size selection: (i) control theory (to keep the error in check); (ii) signal processing (to produce smooth step size sequences and improve computational stability); and (iii) adaptivity, in the sense that the time step should be covariant or contravariant with some prescribed function of the dynamical system’s solution. Examples are used to demonstrate the different advantages in different applications. The main ideas are further developed to approach some open problems that are subject to special requirements.

The numerical behavior of two implicit time-marching methods is investigated in solving two-dimensional unsteady compressible flows. The two methods are the second-order multistep backward differencing formula and the fourth-order multistage explicit first stage, single-diagonal coefficient, diagonally implicit Runge-Kutta scheme. A Newton-Krylov method is used to solve the nonlinear problem arising from the implicit temporal discretization. The methods are studied for two test cases: laminar flow over a cylinder and turbulent flow over a NACA0012 airfoil with a blunt trailing edge. Parameter studies show that the subiteration termination criterion plays a major role in the efficiency of time-marching methods. Efficiency studies show that when only modest global accuracy is needed, the second-order method is preferred. The fourth-order method is more efficient when high accuracy is required. The Newton-Krylov method is seen to be an efficient choice for implicit time-accurate computations. I.

Implicit methods are well known to have greater stability than explicit methods for stiff systems, but they often are not used in practice due to perceived computational complexity. This paper applies the Backward Euler method and a secondorder one-step two-stage composite backward differentiation formula (C-BDF2) for the monodomain equations arising from mathematically modeling the electrical activity of the heart. The C-BDF2 scheme is an L-stable implicit time integration method and easily implementable. It uses the simplest Forward Euler and Backward Euler methods as fundamental building blocks. The nonlinear system resulting from application of the Backward Euler method for the monodomain equations is solved for the first time by a nonlinear elimination method, which eliminates local and non-symmetric components by using a Jacobian-free Newton solver, called Newton-Krylov solver. Unlike other fully implicit methods proposed for the monodomain equations in the literature, the Jacobian of the global system after the nonlinear elimination has much smaller size, is symmetric and possibly positive definite, which can be solved efficiently by standard optimal solvers. Numerical results are presented demonstrating that the C-BDF2 scheme can yield accurate results with less CPU times than explicit methods for both a single patch and spatially extended domains.

Abstract. Monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution. We formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations, and propose an efficient algorithm for its solution. This algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and/or stages. The optimality of some recently proposed methods is verified, and many more efficient methods are found. We use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type, including methods for hyperbolic PDEs that use downwind-biased operators. 1.

Abstract. Planning high-quality camera motions is a challenging problem for applications dealing with interactive virtual environments. This challenge is caused by conflicting requirements. On the one hand we need good motions, formed by trajectories that are collision-free and keep the character that is being followed in clear view. On the other hand, we need frame coherence, i.e. the view must change smoothly such that the viewer does not get disoriented. Since camera motions dynamically evolve, good motions may require the camera to jump, leading to a broken frame coherence. Hence, a careful trade-off must be made. In addition to this challenge, interactive applications require real-time computations, preventing an exhaustive search for ‘the best ’ solution. We propose a new method for planning camera motions which tackles this trade-off in real-time. The method can be used for planning camera motions of npc’s and first-person characters. Experiments show that high-quality camera motions are obtained for both scenarios in real-time. 1

In the framework of the FLITE 2 project, identification algorithms were developed in order to monitor the evolution of the aeroelastic modes of an aircraft during flutter flight tests. Simulated data were required to evaluate and compare the algorithms and also to test their ability to detect the onset of flutter. This paper describes how this simulation was carried out. It presents the model used to simulate the aeroelastic behaviour of the aircraft. A realistic simulation of the in-flight disturbances was also needed to evaluate the identification algorithms in conditions similar to operational conditions. The modeling and simulation of these perturbations constitute another essential feature of this paper. The last point of the article is devoted to the design of appropriate excitation signals. 1