Numerical methods for systems of highly oscillatory ordinary differential equations

The second method, we call it the WRF method, a combination of the Filon method and the waveform relaxation methods, for solving highly oscillating nonlinear systems. The key problem in our discussion throughout the thesis is a family of strongly oscillating vector-valued integrals of the form. In Chapter 2, we briefly describe univariate quadrature methods used to approximate strongly oscillating Fourier-type integrals of the form

In Chapter 3, we present numerical solvers for strongly oscillating linear systems of ODEs with a constant matrix. Developing ideas on the stationary phase approximation [Ste93], we present the asymptotic and Filon methods for solving linear systems of strongly oscillating ODEs. The special combination of the Filon methods and iterative waveform relaxation methods form a new WRF method used to solve highly oscillatory nonlinear systems of ODEs.

We use the FM method for linear systems with a time-dependent matrix and develop the WRFM method, a combination of the WRF and FM methods for nonlinear systems.

Classical quadrature rules

And finally, the formula for the integrand may be known, but it may be difficult or impossible to find the antiderivative, which is an elementary function. The integral of the error function cannot be evaluated in closed form with respect to the elementary functions. However, by expanding the integrand e−z2 into its Taylor series and integrating term by term, we obtain the Taylor series of the error function.

The theory of this method shows that Simpson's rule is exact if the integrand is a polynomial of degree three or lower. The error introduced by the compound Simpson's rule is bounded in absolute value by. In practice, it is often useful to use subintervals of different lengths and focus on places where the integrand behaves worse.

Gauss's n-point quadrature rule is a quadrature rule that is constructed to give an exact result for polynomials of degree 2n−1 or less with appropriate choice of pointsxi and weights wi for i= 1, .., n. The formula.

Highly oscillatory quadrature

Exponential integrators

Before introducing the framework for exponential integrators, we examine several well-known extensions of the Euler method. This method is of order two, for general problems of the form f(y(t)), and exact for problems where. Nørsett in [IN05] that the standard numerical approach based on the Gauss-Christoffel quadrature fails to approximate highly oscillating integrals, since the approximation error is O(1) for ω → ∞. Instead, the authors developed the asymptotic and Filon methods, which share the property that the accuracy improves as ω increases.

To present our readers' motivation for our research, we now briefly state the two important theorems from [IN05] describing the quadrature methods used to approximate a more general family of Fourier-type strongly oscillating integrals of the form Assume that f and g are differentiable up to a sufficiently high order and g is free of stationary points on [a, b]. In that case, we can obtain the asymptotic expansion for the Fourier-type integral. The truncated asymptotic expansion appears to be a powerful tool for the approximation of integralI[f] compared to aforementioned classical methods.

Once the assumption on the differentiation of the functions is satisfied, the asymptotic method uses very little information about the values ​​of the function and the derivatives at the endpoints.

The univariate Filon-type method

In Figure 2.2.1 we present numerical results for the asymptotic and Filon methods with function values ​​and its first derivative only at the endpoints, c1 = 0, c2 = 1, for the integral. However, as we can see from Figure 2.2.1, the Filon method provides a greater measure of accuracy than the asymptotic method. We would like to emphasize that the Filon method also works for small values ​​of ω.

With Gauss points, it is equivalent to Gauss-Christoffl quadrature for ω → 0 using the same information. In [IN05], the authors showed that adding more interior points causes the leading error constant to drop, resulting in a noticeable improvement in the approximation accuracy.

Applications to linear systems

For simplicity, consider a spectral decomposition of the matrix Aω =P DP−1, with a purely imaginary spectrum σ(Aω) ={iωk}dk=1, ωk∈R,. This suggests that to approximate a strongly oscillatory integral I[f] with a matrix-valued kernel and a vector-valued function in (2.0.1), we need a strongly oscillatory integral of the kind (2.0.2) in I[ to approach. F]. Needless to say, the asymptotic order of our approximation of I[f] depends on the inverse powers of the eigenvalues, so we want the error to decrease as the eigenvalues ​​increase.

For smaller eigenvalues, the method is comparable to classical methods, while in the case of zero eigenvalues ​​the method will be equivalent to the polynomial approximation of the integrable function, [Kha08b]. At this point it would be appropriate to introduce some notes on matrix and function asymptotics from [Olv07]. Note that the constant inAB =O(˜AB˜) does not depend on the dimension of the matrices, but on the largest absolute value of a constant in the elementary evaluations of the entries of a matrix product.

Thus, the constant v AB = O( ˜AB˜) depends on the maximum value ˜C = max{C˜ij}, which will provide a uniform estimate for ∀kzijk ≤C˜kz˜ijka, and not on the dimension of the matrices.

The Filon-type method for matrix-valued kernels

Using the initial value integrator (5.1.3), we present the Filon method for the systems with strongly oscillating ODEs. Then r is the numerical order of the Filon method applied to the linear system. We would like to remind our reader that σ(Aω)⊂iR, which means that the norm of the matrix exponential is always bounded, etJ = O(1).

In other words, for a fixed value of parameter ω, the convergence of the numerical scheme to the exact solution follows from the estimation of the residual term. Comparison will show that the Filon type method outperforms the asymptotic method even though both methods are of the same asymptotic order. The logarithmic error in Figure 3.2.4 describes both numerical and asymptotic analysis of the ω increasing method.

It follows from the spectral decomposition of the matrix Aω that the factor eiλkf appears in the fundamental. The new method, called the FM method, combines the Magnus method, which is based on iterative techniques to approximate Ω, and the Filon-type method, an efficient quadrature rule for solving oscillating integrals. We make the following assumptions: Aω(t) is a smooth matrix-valued function, the spectrum σ(Aω) of the matrix Aω has large imaginary eigenvalues ​​and that ω ≫1 is a real parameter describing the frequency of oscillation.

Hausdorff [Hau06] showed that the solution of the linear matrix equation in (4.1.1) is the matrix exponential Xω(t) = eΩ(t), where Ω(t) satisfies the following nonlinear differential equation,. Magnus methods preserve the special geometric properties of the solution and ensure that if Xω is in the matrix Lie group G and Aω is in the connected Lie algebra g of G, the numerical solution remains on the manifold after discretization. Using the Magnus method for solving highly oscillating differential equations, we developed it in combination with appropriate quadrature rules, such as Filon's methods, which have been shown to be more efficient for oscillating equations than, for example, classical Gaussian quadrature.

In the following, we remind our reader of Philon's method and the asymptotic method, which are used for the approximation of highly oscillating integrals of the form. For a formal proof of the asymptotic order of the Filon-type method for univariate integrals, we refer the reader to [IN05] and for matrix and.

The Magnus method

In the paper, authors provide asymptotically sharp error bounds for Magnus integrators in the framework applied to time-dependent Schr¨odinger equation where there are no constraints on smallness or bounds onhkH(t)k, h stands for a step size of a numerical integrator. Comparison shows that for a larger interval steps hboth fourth and sixth order methods give similar results, as illustrated in Figures and 4.2.6. However, for smaller steps six-order Magnus method has a faster improvement in approximation compared to a fourth-order method, Figures 4.2.3 and 4.2.6.

In present work, we present an alternative method to solve equations of the kind (4.1.1), [Kha09a]. We apply the Filon quadrature to evaluate integral commutators for Ω and then solve the matrix exponential Xω using the Magnus method or the modified Magnus method. The combination of the Filon-type methods and the Magnus methods forms the FM method, which is presented in the next section.

The implementation of the FM method for solving multi-oscillator ordinary differential equation systems can be found in [Kha09b], [Kha09a].

The modified Magnus method

The FM method

Here, the integral commutators in ˜Ωn are calculated according to the rules for the Filon quadrature. In the example below, we provide a more detailed evaluation of the FM method used to solve the Airy-type equation y′ =Aω(t)y. It is possible to reduce the cost of evaluating the matrix B(t) by simplifying it, [Ise04a].

Given the compact representation of the matrix B(t) with oscillatory entries, we solve Ω(t) using Filon's method. In Figure 4.4.1 we present the global error of the fourth-order modified Magnus method with exact integrals for the Airy equation y′′(t) = −ty(t) with [1,0]⊺ initial conditions, 0 ≤t≤ 2000 time interval and size steps h= 15. Then r is the numerical order of Philo's method applied to linear systems.

Applications of the modified Magnus method include systems of highly oscillating linear ODE systems, [Kha09a]. The cost of evaluating the matrix B(t) can be reduced by simplification, [Ise04a]. Given the compact representation of the matrix B(t) with oscillatory entries, we solve Ω(t) using the Filon-type method, approximating the function.

Performance of the modified Magnus method improves compared to the classical Magnus method due to a number of reasons. It was also proved in [Van93] that the iteration is contraction for the derivatives of the iterate, . In approximating highly oscillating nonlinear systems of ODEs, we use the implicit representation of the solution and the initial value integrator.

The computational cost of the WRF method is comparable to that of the Filon-type method. In Table 6.2.1 we demonstrate the accuracy of the WRF method for increasing ω in just four iterations. Further applications of the Modified Magnus method include systems of highly oscillatory nonlinear equations.

The modified WRFM method is defined as a local approximation of the solution vector y by solving.