The FM method - Numerical methods for systems of highly oscillatory ordinary differential equat

0 200 400 600 800 1000

−4

−2 0 2

4 x 10

⁻³

Figure 4.2.4: Global error of the six order Magnus method for the Airy equation y^′′(t) = −ty(t) with [1,0]^⊺ initial conditions, 0 ≤ t ≤ 1000 and step-size h= ¹₄.

Performance of the modified Magnus method is better than that of the classicalMagnusmethod due to a number of reasons. Firstly, the fact that the matrix B is small, B(t) =O(t−t_n+¹

2),contributes to higher order correction to the solution. Secondly, the order of themodified Magnusmethod increases from p= 2s to p= 3s+ 1, [Ise02b].

4.4 The FM method

In this section we present a novel FM method for solving matrix ODEs, [Kha09a]. The method combines ideas on Filon methods and Lie group methods for solving matrix exponential with nested integral commutators.

Consider theAiry-type equation

y^′′(t) +g(t)y(t) = 0, g(t)>0 for t >0, and lim

t→∞g(t) = +∞. Replacing the second order differential equation by the first order system, we

4.4 The FM method

0 200 400 600 800 1000

−8

−4 0 4

8 x 10

⁻⁴

Figure 4.2.5: Global error of the six order Magnus method for the Airy equation y^′′(t) = −ty(t) with [1,0]^⊺ initial conditions, 0 ≤ t ≤ 1000 and step-size h= ₁₀¹.

obtain y^′ =Aω(t)y, where

A_ω(t) =

0 1

−g(t) 0

Due to a large imaginary spectrum of the matrixA_ω, thisAiry-type equation is rapidly oscillating. We apply the modified Magnus method to solve the system y^′ =Aω(t)y,

y_n+1 = e^hA^ω^(tⁿ^+h/2)e^Ω^˜ⁿy_n.

Here the integral commutators in ˜Ωn are computed according to the rules of the Filon quadrature. This results in highly accurate numerical method, the FM method. Taking into account that the entries of the matrixB(t) are likely to be oscillatory, the advantage of the Filon-type method is evident.

In the example below we provide a more detailed evaluation of the FM method applied to solve the Airy-type equation y^′ =Aω(t)y.

4.4 The FM method

0 200 400 600 800 1000

−5 0

5 x 10

⁻⁹

Figure 4.2.6: Global error of the six order Magnus method for the Airy equation y^′′(t) = −ty(t) with [1,0]^⊺ initial conditions, 0 ≤ t ≤ 1000 and step-size h= ₂₅¹.

Example 4.4.1 Once we have obtained the representation for Ω˜n, the com- mutator brackets are now formed by the matrix B(t). It is possible to reduce cost of evaluation of the matrix B(t) by simplifying it, [Ise04a]. Denote q =q

g(tn+¹₂h) and v(t) =g(tn+t)−g(tn+¹₂h). Then, B(t) =v(t)

(2q)⁻¹sin 2qt q⁻²sin²qt

−cos²qt −(2q)⁻¹sin 2qt

=v(t)

q⁻¹sinqt

−cosqt

cosqt q⁻¹sinqt , and for the product

B(t)B(s) =v(t)v(s)sinq(s−t) q

q⁻¹sinqt

−cosqt

cosqs q⁻¹sinqs . It was shown in [Ise04a] that

kB(t)k= cos²ωt+sin²ωt

ω² and B(t) =O(t−t_n+¹

2).

4.4 The FM method

We can now estimate integral Zh

kB(t)kdt= h

2(1 + 1

ω²) + sin 2ωh

2ω (1− 1

ω²) := q(h, ω).

To meet requirements stated in Theorem 4.2.1 we allow q(h, ω)≤π.

In other words, to satisfy Theorem 4.2.1 we obtain the following condition on the step-size h,

h≤ 2πω³−ω²+ 1 ω(ω²+ 1) .

Given the compact representation of the matrix B(t) with oscillatory entries, we solve Ω(t) with a Filon-type method. We approximate functions in B(t) by a polynomial ˜v(t), for example Hermite polynomial (4.1.4), as in classical Filon-type method. Furthermore, in our approximation we use end points only, although the method is general and more nodes of approximation can be required.

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 step-size h= ¹₅. This can be compared with the global error of the FMmethod applied to the same equation with exactly the same conditions and step-size, Figure 4.4.2.

In Figures 4.4.3 and 4.4.4 we compare the fourth order classical Magnus method with a remarkable performance of the fourth order FM method, applied to the Airy equation with a large step-size equal to h= ¹₂. While in Figures 4.4.5 and 4.4.6 we show the solution for the Airyequation with the fourth order FM method with step-sizes h= ¹₄ and h= ₁₀¹ respectively.

4.4 The FM method

0 500 1000 1500 2000

−3

−2

−1 0 1 2

x 10

⁻¹⁰

Figure 4.4.1: Global error of the fourth order Modified Magnus method with exact evaluation of integral commutators for the Airy equationy^′′(t) =−ty(t) with [1,0]^⊺ initial conditions, 0≤t≤2000 and step-size h= ¹₅.

0 500 1000 1500 2000

−3

−2

−1 0 1 2

x 10

⁻¹⁰

Figure 4.4.2: Global error of the fourth order FM method with end points only and multiplicities all 2 for the Airy equation y^′′(t) =−ty(t) with [1,0]^⊺ initial conditions, 0 ≤t≤2000 and step-size h = ¹₅.

4.4 The FM method

0 500 1000 1500 2000

−4

−2 0 2 4

x 10

⁻⁷

Figure 4.4.3: Global error of the fourth order FM method with end points only and multiplicities all 2 for the Airy equation y^′′(t) =−ty(t) with [1,0]^⊺ initial conditions, 0 ≤t≤2000 and step-size h = ¹₂.

0 500 1000 1500 2000

−0.015

−0.005 0.005 0.015

Figure 4.4.4: Global error of the fourth order Magnus method for the Airy equation y^′′(t) = −ty(t) with [1,0]^⊺ initial conditions, 0 ≤ t ≤ 2000 and step-size h= ¹₂.

4.4 The FM method

0 500 1000 1500 2000

−1 0 1

x 10

⁻⁹

Figure 4.4.5: Global error of the fourth order FM method with end points only and multiplicities all 2, for the Airy equation y^′′(t) =−ty(t) with [1,0]^⊺ initial conditions, 0 ≤t≤2000 and step-size h = ¹₄.

0 500 1000 1500 2000

−4

−2 0 2

4 x 10

⁻¹¹

Figure 4.4.6: Global error of the fourth order FM method with end points only and multiplicities all 2,for the Airy equation y^′′(t) =−ty(t) with [1,0]^⊺ initial conditions, 0 ≤t≤2000 and step-size h = ₁₀¹.

Chapter 5 Highly oscillatory linear systems with variable coefficients

5.1 The asymptotic method

In this chapter we introduce our reader further applications of the FM methods. In particular, we develop the FM method for solving highly oscillatory systems of ODEs with a time-dependent matrix, [Kha09b], [Kha09a]. We solve the matrix exponential in Lie groups applying Magnus method. For discretizing nested commutators we apply Filon quadrature and develop the FM method.

We commence from a system of ordinary differential equations written in a vector form,

y^′ =Aωy+f(t), y(0) =y₀ ∈R^d, t≥0,

whereAω is a non-singular matrix with large imaginary eigenvalues,σ(Aω)⊂ iR, kAωk ≫ 1,ω ≫1 is a real parameter describing frequency of oscillation and f :R×R^d→R^d is a smooth vector-valued function.

In present chapter we allow matrixAωto be time-dependent non-singular matrix as compared to a constant matrix Aω, discussed previously and presented in [Kha08b]. For more generality one can assume that Aω can be a function from [0,∞) to the set of matrices. However, the scope of this

5.1 The asymptotic method

thesis is to provide answers while Aω is the source of the oscillation occur- ring in the system and the methods introduced in our work are beneficial for large frequencies of oscillation. While for small frequencies or even in absence of oscillation our discretization methods are of the same order as Gauss-Christopher rule with Gaussian point.

The analytic solution of the system is given by the variation of constants formula,

y(t) =Xω(t)y0+ Z t

Xω(t−τ)f(τ,y(τ))dτ =Xωy₀+I[f]. (5.1.1) The fact that Aω is no longer a constant matrix results in a more complex solution of a fundamental matrix Xω, satisfying matrix linear ODE,

X_ω^′ =A_ωX_ω, X_ω = e^Ω,

where Ω represents Magnus expansion, an infinite recursive series, Ω(t) =

Z t 0

A(t)dt +1

2 Z t

[A(τ), Z τ

A(ξ)dξ]dτ +1

4 Z t

[A(τ), Z τ

[A(ξ), Z ξ

A(ζ)dζ]dξ]dτ + 1

12 Z t

[[A(τ), Z τ

A(ξ)dξ], Z τ

A(ζ)dζ]dτ +· · · .

The challenge is to evaluate highly oscillatory integral I[f] in (5.1.1) effi- ciently. We already know that ordinary methods based onGauss–Christoffel quadrature fail to approximate highly oscillatory integrals for increasing ω, [IN05]. Hence, we first focus on developing numerical methods to evaluate highly oscillatory integrals with a matrix-valued kernel and next we use our results in numerical approximation to (5.1.1).

Given a vector-valued integral over a compact interval [a, b]

I[f] = Z b

Xω(t)f(t)dt, X_ω^′ =AωXω, Xω = e^Ω,

5.1 The asymptotic method

with an arbitrary non-singular matrix Aω of large eigenvalues, kA⁻_ω¹k ≪ 1, σ(Aω)⊂iR, a real parameterωand a smooth vector-valued functionf ∈R^d. We would like to remind our reader that if the matrix Aω was constant, then for example the task of obtaining an asymptotic expansion for the integral I[f] was quite simple, [Kha08b]. Applying rules of integration by parts,

I[f] =A⁻_ω¹

Xω(b)f(b)−Xω(a)f(a)

−A⁻_ω¹ Z b

Xω(t)f^′(t)dt

=Q^A₁ −A⁻_ω¹I[f^′],

we obtain theasymptotic methodQ^A_s, defined ass-partial sum of the asymptotic expansion for I[f],

Q^A_s[f] =−

m=1

(−Aω)⁻^mh

Xω(b)f^(m⁻¹⁾(b)−Xω(a)f^(m⁻¹⁾(a)i .

This is not the case when the matrix Aω is arbitrary, since substitution Xω =A⁻_ω¹X_ω^′ inI[f] is no longer useful to apply tools of integration by parts,

I[f] = Z b

A⁻_ω¹(t)X_ω^′(t)f(t)dt.

However, it is possible to overcome this issue and obtain the asymptotic expansion for I[f] given an arbitrary matrixAω.

In this section we show how applying some matrix transformations inI[f] leads to its explicit asymptotic expansion.

Given, I[f] =

Z b a

X_ω(t)f(t)dt, X_ω^′ =A_ωX_ω, X_ω = e^Ω, (5.1.2) with a matrix valued kernel Xω depending on a real parameter ω ≫ 1 and satisfying matrix linear differential equation (5.1.2). Assume that A_ω is an arbitrary non-singular matrix of pure imaginary spectrum, σ(Aω)⊂iR, kA⁻_ω¹k ≪1 and f ∈R^d is a smooth vector-valued function.

We have already mentioned previously in this chapter that the current representation of the integral I[f] in (5.1.2) is not suitable for obtaining its

5.1 The asymptotic method

asymptotic expansion due to a more general choice of the matrix Aω. Once we substitute Xω =A⁻_ω¹X_ω^′ in I[f] in its current occurrence, then

I[f] = Z b

A⁻_ω¹(t)X_ω^′(t)f(t)dt.

Transforming integral (5.1.2) into an equivalent integral with appropriate representation for our purposes helps to obtain the asymptotic expansion to (5.1.2).

We first vectorize matrixX_ω into

X_ω = [¯x₁,x¯₂, ...,x¯_d].

Next step we take the transpose of X_ω, obtaining a corresponding vector X¯_ω,

X¯_ω = [x¯₁,x¯₂, ...,x¯_d]^⊺.

It is easy to verify that the latter satisfies a linear differential equation X¯^′_ω =BωX¯_ω, with Bω =

i=1

Aω,

where Bω represents direct d-tuple sum of the original matrixAω.

We also scale an identity matrix I by taking its direct product with the vector-valued function f := f^⊺ = [f1, f2, . . . , fd], F = f₁_×_dN

Id×d. For instance, take

f = [f1, f2] and I =

1 0 0 1

. Direct product F =fN

I results in the following matrix, F =

f1 0 f2 0 0 f1 0 f2

. Thus, F =

i=1

Fi, where each entry matrix is a diagonal matrix with the corresponding element fi on the diagonal,

Fi =







fi 0 . . . 0 0 fi . . . 0

0 0 fi 0

... . .. ... ...

0 . . . 0 fi





 .

5.1 The asymptotic method

These transformations result in equalityXωf =FX¯_ω,and appear to be suitable for integration by parts and representation of the asymptotic expansion of the integral (5.1.2).

Example 5.1.1 Let Xω =

x11 x12

x₂₁ x₂₂

, Aω =

a11 a12

a₂₁ a₂₂

and f = f1

f₂

. Then

X¯_ω =





 x11

x₂₁ x12

x22







, B_ω =







a11 a12 0 0 a₂₁ a₂₂ 0 0 0 0 a11 a12

0 0 a21 a22







and

F =

f1 0 f2 0 0 f1 0 f2

. Hence,

Xωf =FX¯_ω =

x11f1+x12f2

x21f1+x22f2

, and X¯^′_ω =BωX¯_ω, once X_ω^′ =AωXω.

As a result, we have obtained the following equality for the two integrals, I[f] =

Z b a

X_ω(t)f(t)dt= Z b

F(t)X¯_ω(t)dt=I[F], where matrix Xω and vectorX¯_ω satisfy linear ODEs

X_ω^′ =AωXω and X¯^′_ω =BωX¯_ω

respectively. This transformation allows us to approximate I[F] and thus I[f]. Applying integration by parts toI[F] we obtain the asymptotic expansion for the integral I[f],

I[f] = Z b

F(t)X¯_ω(t)dt= Z b

F(t)B_ω⁻¹(t)X¯^′_ω(t)dt

F(t)B_ω⁻¹(t)X¯_ω(t)b a−

Z b a

F(t)B_ω⁻¹(t)_′X¯_ω(t)dt.

5.1 The asymptotic method

Hereafter we remind our reader the notation on matrix functions asymp- totics depending on a real parameter ω. For the entirety of our work, all norms are L^∞ norms, for vectors, matrices and functions. The norm of a function is taken over the interval [a, b]. We say that f = O( ˜f) for an arbitrary function f and non-negative constant ˜f, which depend on a real parameter ω, if the norm of f and its derivatives are all of order O( ˜f) as ω → ∞, namely kf^(m)k = O( ˜f) for m = 0,1, . . .. For arbitrary two n×m matrices A(x) = (aij(x)) and Ã = (ãij), ãij ≥ 0, depending on a real parameter ω, we can thus posit A(x) = O( Ã), if aij(x) = O(ãij) element-wise as ω → ∞. We may also say that f = O(1), if f and its derivatives remain bounded on [a, b], as ω → ∞. Let 1 ={1ij} stand for a matrix with all entries one. This allows us to write A(x) =O(1), ifaij(x) =O(1) element-wise as ω → ∞. And finally, if A = O( Ã) and B = O( ˜B), then the integration and multiplication properties are Rb

a A(x)dx =O( ˜A) and AB =O( ˜AB).˜ Letting

σ₀=F,

σk+1= (σkB_ω⁻¹)^′, k= 0,1, . . . we define asymptotic method as

Q^A_s[f] =

s−1

k=0

(−1)^k σ_k(b)B_ω⁻¹(b)X¯_ω(b)−σ_k(a)B⁻_ω¹(a)X¯_ω(a) .

Below we Theorem 3.1 from [Olv07] and Lemma 2.1 from [Kha08b]. In Theorem 5.1.3 we generalize these results for a matrix-valued kernel Xω, a vector-valued function f and a time dependant matrix Aω inI[f], (5.1.2) . Theorem 5.1.1 (S. Olver, [Olv07]) Suppose that y satisfies the differential equation,

y^′(x) =A(x)y(x),

in the interval [a, b], for some invertible matrix-valued function A such that A⁻¹ =O( ˆA), for ω → ∞. Assume that

I[f] = Z b

f^⊺(x)y(x)dx,

5.1 The asymptotic method

where f ∈ R^d is a smooth vector-valued function and y ∈ R^d is a smooth, highly oscillatory vector- valued function. Define

Q^A_s[f] =

s−1

k=0

(−1)^k[σ_k^⊺(b)A⁻¹(b)y(b)−σ_k^⊺(a)A⁻¹(a)y(a)], where

σ0 ≡f, σk+1 = (A⁻^⊺σk)^′, k = 0,1, . . . . If f =O( ˜f) and y(x) =O(y)˜ for a≤x≤b, then

I[f]−Q^A_s[f] = (−1)^s Z b

σ_s^⊺ydx=O(f˜^⊺Aˆ^s+1y),˜ as ω → ∞. Lemma 5.1.2 (MK, [Kha08b]) Let

I[f] = Z b

Xω(t)f(t)dt, X_ω^′ =AωXω,

where the matrix kernel Xω satisfies linear matrix ODE as above, Aω is a constant non-singular matrix, kA⁻_ω¹k ≪ 1 and f : R → R^d is a smooth vector-valued function. Then, for ω≫1,

I[f]∼ − X∞

m=1

(−Aω)⁻^m

Xω(b)f^(m⁻¹⁾(b)−Xω(a)f^(m⁻¹⁾(a) .

For ψ = max{kf^(s)k,kf^(s+1)k},

Q^A_s[f]−I[f]∼O(kA⁻_ω^s⁻¹kkXωkψ), as ω→ ∞. If X_ω =O( ˆX_ω) and f =O(f˜), then

Q^A_s[f]−I[f] = O(A⁻_ω^s⁻¹Xˆωf˜), as ω → ∞, element wise.

5.1 The asymptotic method

Theorem 5.1.3 [Kha09b] Postulate that I[f] =

Z b a

X_ω(t)f(t)dt, and X_ω^′ =A_ω(t)X_ω,

whereAω is an arbitrary non-singular matrix,σ(Aω)⊂iR,ωis an oscillatory parameter and f ∈R^d is a smooth vector-valued function.

If F_ω =O(Fˆ), B⁻_ω¹ =O( ˆB) and X¯_ω =O(Xˆ_ω), then

I[f]−Q^A_s[f] = (−1)^s Z b

σ_s(t)X¯ω(t)dt

= O(FˆBˆ^s+1X),ˆ as ω → ∞. Proof: We first prove the identity by induction,

I[F] =

F(t)B_ω⁻¹(t)X¯_ω(t)b a−

Z b a

F(t)B_ω⁻¹(t)′X¯_ω(t)dt

σ0(t)B_ω⁻¹(t)X¯_ω(t)b a−

Z b a

σ1(t)X¯_ω(t)dt.

Suppose the identity holds for some k ∈N, we prove fork+ 1.

Z b a

σk+1(t)X¯_ω(t)dt= Z b

σk+1(t)B_ω⁻¹(t)X¯^′_ω(t)dt

σ_k+1(t)B_ω⁻¹(t)X¯_ω(t)b a−

Z b a

σ_k+2(t)X¯_ω(t)dt.

By induction, σk =O(FˆBˆ^k). Indeed, fork = 0,σ0 =F =O(Fˆ). We assume the equality holds for some σk, then

σk+1=

σkB_ω⁻¹_′

=σ^′_kB_ω⁻¹+σkB_ω⁻¹^′

=O(FˆBˆ^k)O( ˆB) +O(FˆBˆ^k)O( ˆB) =O(FˆBˆ^k+1).

And finally, Z b

σsX¯_ωdt=

σsB_ω⁻¹X¯_ωb a−

Z b a

σs+1X¯_ωdt

= O(FˆBˆ^s+1X) + O(ˆ FˆBˆ^s+1X) = O(ˆ FˆBˆ^s+1Xˆ).

5.1 The asymptotic method

Corollary 5.1.4 [Kha09b] If

f^(k)(a) =f^(k)(b) = 0, for k = 0,1, ..., s−1, then,

I[f] = O(FˆBˆ^s+1Xˆ).

Example 5.1.2 For practical purposes consider integral

I_h[f] = Z h

e^Ω(h⁻^τ)f(τ)dτ.

Once we vectorize the system to obtain asymptotic method, I¯stands for a vectorized identity matrix I, andX¯_ω stands for a vectorized matrix e^Ω. For s = 2 the asymptotic method using information at the end points only, is

Q^A₂ = [F(h)B_ω⁻¹(0)−F^′(h)B_ω⁻²(0)−F(h)B_ω⁻¹^′(0)B_ω⁻¹(0)]I¯

−[F(0)B_ω⁻¹(h)−F^′(0)B_ω⁻²(h)−F(0)B_ω⁻¹^′(h)B⁻_ω¹(h)]X¯_ω(h).

Employing initial-value integrator, y_n+1 = e^Ω^sy_n+

Z h 0

e^Ω^s^(h⁻^τ)f(tn+τ)dτ,

where Ωs stands for a truncatedMagnus expansion, we can now solve (5.1.1) with the asymptoticmethod,

y_n+1 = e^Ω^sy_n+Q^A_s.

We solve matrix exponential e^Ω^s with theFMmethod described in [Kha09a].

The method combines the ideas ofLie groupsolvers, namelyMagnusmethod, with Filon quadrature for solving nested integral commutators in Ω.

The following lemma for analysis of highly oscillatory ODEs instead of Taylor method, present a paragraph on it

Lemma 5.1.5 [Kha09a] The approximation with the asymptotic method Q^A_s applied to an integral I[f], is independent of the step-size of integration.

5.2 The FM method

Proof: For∀step-sizehand∀numbernof node pointstn =tn−1+h, the components evaluated at the internal node points of the truncated asymptotic sum Q^A_s appear in pairs with an opposite sign, except the values at the end points

QÂ_s[f](t1)−QÂ_s[f](t0) +QÂ_s[f](t2)−QÂ_s[f](t1) +QÂ_s[f](t3)−QÂ_s[f](t2) +...

+QÂ_s[f](tn−1)−QÂ_s[f](tn−2) +QÂ_s[f](tn)−QÂ_s[f](tn−1)

=Q^A_s[f](tn)−Q^A_s[f](t0).

As a result, no matter how we truncate the interval we obtain the same representation for the asymptotic method Q^A_s evaluated at the end points.

5.2 The FM method

Employing classical Filon-type quadrature proven to have high accuracy for increasing ω, as in [Kha08a], we construct anr-degree polynomial interpola- tion v for the vector-valued function f in (5.1.2),

v(t) =

l=1 θl−1

j=0

αl,j(t)f^(j)(tl),

which agrees with function values and derivatives v^(j)(tl) =f^(j)(tl) at node points a =t1 < t2 < ... < tν = b, with associated θ1, θ2, ..., θν multiplicities, j = 0,1, ..., θ_l−1, andl = 1,2, ..., ν, [Kha09b].

By definition, Filon-type method is Q^F_s[f] =

Z b a

Xω(t)v(t)dt=

l=1 θl−1

j=0

βl,jf^(j)(tl),

where βl,j =Rb

a Xω(t)αl,j(t)dt.

Theorem 5.2.1 [Kha09b] Postulate that

I[f] = Z b

Xω(t)f(t)dt, and X_ω^′ =Aω(t)Xω,

5.2 The FM method

Aω is a non-singular matrix, σ(Aω) ⊂ iR, ω is an oscillatory parameter and f ∈R^d is a smooth vector-valued function.

If F_ω =O(Fˆ), B⁻_ω¹ =O( ˆB) and X¯_ω = O(Xˆ_ω), then

I[f]−Q^F_s[f] = (−1)^s Z b

σs(t)X¯ω(t)dt

= O(FˆBˆ^s+1X),ˆ as ω → ∞.

Proof: Observing that I[f −v] = I[f]−Q^F_s[f], the proof follows by replacing f in the asymptotic method (5.1.3) by f −v and the fact that [f −v]^(j)(a) =[f −v]^(j)(b) = 0, for j = 0,1, ..., s−1.

Corollary 5.2.2 [Kha09b] If Xω =O(1) and f =O(1), then Q^F_s[f]−I[f] = O(A⁻_ω^s⁻¹1).

Theorem 5.2.3 [Kha09b] Let θ₁, θ_ν ≥ s, r = Pν

l=1θ_l −1. Then r is the numerical order of the Filon-type method applied to the linear systems,

y(tn)−y_n =O(h^r+1).

Numerical experiments withQ^{F M}_s method, a combination ofFilonquadra- ture and Magnus method, are available in Figures 5.2.1, 5.2.2, 5.2.3, 5.2.4, 5.2.5, 5.2.6 and 5.2.7.

By definition, FM method is Q^{F M}_s [f] =

Z b a

e^Ω^s^(t)v(t)dt=

l=1 θl−1

j=0

βl,jf^(j)(tl),

where β_l,j = Rb

a e^Ω^s(t)α_l,j(t)dt. In our case it is numerical discretization of the integral

I[f] = Z h

e^Ω^sf(tn+τ)dτ,

5.3 The modified FM method

applied to solve

y_n+1 = e^Ω^sy_n+Q^{F M}_s [f].

We present numerical results forFM method Q^{F M}₂ with end points only and multiplicities all 2, for the equation y^′′(t) =−ty(t)−cos(t), 0≤t≤100, with [1,0]^⊺ initial conditions and various step-sizes in Figures 5.2.1-5.2.7.

Initially, we show that employing the FM method one can afford large step- sizes, such ash= 1 andh= ¹₂,Figures 5.2.1. Decreasing step-size up toh= ¹₄ and h= ₁₀¹ we obtain accuracy up to 10⁻⁸, Figure 5.2.2. In Figure 5.2.4 we show that for a fixed large step-size the accuracy of approximation improves with an increase in frequency,h= ¹₄, ω= 10²and 10³.For a smaller step-sizes we demonstrate high accuracy in approximation for ω = 10⁴, Figure 5.2.5 and ω = 10⁵, Figure 5.2.6. While in Figure 5.2.7 we show that for ω = 10⁶ and hω = 10⁴ the method introduces highly desirable accuracy, the error is 10⁻¹⁴. This can be compared with Figure 5.2.3 for the same step-size size.

5.3 The modified FM method

In our further discussion we develop a modified FM method Q^{F M}_S employing ideas from the modified Magnus method, [Ise02c], [Kha09a]. By solving the equation for variable constants locally with a constant matrix we introduce local change of variables. Applications of the modified Magnus method include systems of highly oscillatory linear systems of ODEs, [Kha09a]

y^′(t) =Aω(t)y(t) +f(t), y(t0) =y₀, with analytic solution

y(t) =X_ω(t)y₀+ Z t

X_ω(t−τ)f(τ)dτ =X_ωy₀+I[f].

We assume that the spectrum of the matrixA_ω(t) has large imaginary values.

Introducing local change of variables at each mesh point, we write the solution in the form

y(t) = e^(t⁻^tⁿ^)A(tⁿ⁺¹²^h)x(t−t_n), t ≥t_n.

5.3 The modified FM method

0 20 40 60 80 100

−0.01

−0.005 0 0.005 0.01

0 20 40 60 80 100

−2

−1 0 1 2 x 10

⁻⁴

Figure 5.2.1: Global error of the FM methodQ^{F M}₂ with end points only and multiplicities all 2, for the equation y^′′(t) = −ty(t)−cos(t), 0 ≤ t ≤ 100, with [1,0]^⊺ initial conditions and step-size h= 1 and h= ¹₂ (top two figures from left to right)

We apply modifiedMagnustechniques to the system and solve the system locally with a constant matrix Aω, while x(t) satisfies a highly oscillatory equation

x^′(t) =Bω(t)Xω(t) + e⁻^tA^ω^(tⁿ⁾f(t), t≥0, with the same matrix B_ω as for linear matrix ODEs,

Bω(t) = e⁻^tA^ω^(tⁿ⁾[Aω(t)−Aω(tn)]e^tA^ω^(tⁿ⁾.

It is possible to reduce cost of evaluation of the matrixB(t) by simplifying it, [Ise04a]. Denoteq=q

g(tn+¹₂h) andv(t) = g(tn+t)−g(tn+¹₂h). Then, B(t) =v(t)

(2q)⁻¹sin 2qt q⁻²sin²qt

−cos²qt −(2q)⁻¹sin 2qt

=v(t)

q⁻¹sinqt

−cosqt

cosqt q⁻¹sinqt ,

5.3 The modified FM method

0 20 40 60 80 100

−4

−2 0 2 4 x 10

⁻⁶

0 20 40 60 80 100

−4

−2 0 2 4 x 10

⁻⁸

Figure 5.2.2: Global error of the FM methodQ^{F M}₂ with end points only and multiplicities all 2, for the equation y^′′(t) = −ty(t)−cos(t), 0 ≤ t ≤ 100, with [1,0]^⊺ initial conditions and step-size h = ¹₄ and h = ₁₀¹ (from left to right).

and for the product

B(t)B(s) =v(t)v(s)sinq(s−t) q

q⁻¹sinqt

−cosqt

cosqs q⁻¹sinqs . It was shown in [Ise04a] that

kB(t)k= cos²ωt+sin²ωt

ω² and B(t) =O(t−t_n+¹

2).

This approximation results in the following algorithm, y_n+1= e^hA(tⁿ^+h/2)x_n

x_n= e^Ω^˜ⁿy_n.

Given the compact representation of the matrix B(t) with oscillatory entries, we solve Ω(t) with a Filon-type method, approximating function

5.3 The modified FM method

0 20 40 60 80 100

−4

−2 0 2

4 x 10

⁻¹³

Figure 5.2.3: Global error of the FM methodQ^{F M}₂ with end points only and multiplicities all 2, for the equation y^′′(t) = −ty(t)−cos(t), 0 ≤ t ≤ 100, with [1,0]^⊺ initial conditions and step-size h= ₁₀₀¹ .

v(t) by a polynomial ˜v(t), for example Hermite polynomial, as in classical Filon-type method. Furthermore, in our approximation we use end points only, although the method is general and more nodes of approximation can be required.

As a result we can now solve (5.1.2) we Modified Magnus method in combination with Filon quadrature, which we call the modified FM method, and this proves to be more accurate approximation than solving if we solved the commutators with Gaussian quadrature. Performance of the modified Magnus method improves as compared to classical Magnus method due to a number of reasons. The fact due to the fact that the matrix B is small, B(t) = O(t −t_n+¹

2), contributes to higher order correction to the solution, [Ise02b], [Kha09a]. The order of the modified Magnus method increase from p= 2s top= 3s+ 1.

5.3 The modified FM method

0 20 40 60 80 100

−1.5

−1

−0.5 0 0.5 1 1.5x 10⁻⁶

0 20 40 60 80 100

−2

−1 0 1 2x 10⁻⁷

Figure 5.2.4: Global error of the Q^{F M}₂ FMmethod with end points only and multiplicities all 2, for the equation y^′′(t) = −ωty(t)−cos(t), 0 ≤ t ≤ 100, with [1,0]^⊺ initial conditions and step-size h= ¹₄ and ω= 10² ω = 10³ (from left to right).

0 20 40 60 80 100

−1.5

−1

−0.5 0 0.5 1 1.5x 10⁻⁹

0 20 40 60 80 100

−4

−2 0 2 4x 10⁻¹¹

Figure 5.2.5: Global error of the Q^{F M}₂ FMmethod with end points only and multiplicities all 2, for the equation y^′′(t) = −ωty(t)−cos(t), 0 ≤ t ≤ 100, with [1,0]^⊺ initial conditions, ω = 10⁴, step-sizes h = ₁₀¹ and h = ₂₅¹ (from left to right),

5.3 The modified FM method

0 20 40 60 80 100

−3

−2

−1 0 1 2 3x 10⁻¹²

0 20 40 60 80 100

−6

−4

−2 0 2 4 6x 10⁻¹⁴

Figure 5.2.6: Global error of the Q^{F M}₂ FMmethod with end points only and multiplicities all 2, for the equation y^′′(t) = −ωty(t)−cos(t), 0 ≤ t ≤ 100, with [1,0]^⊺ initial conditions and step-size h = ₁₀₀¹ with ω = 25×10² and ω = 10⁵ (from left to right.)

0 20 40 60 80 100

−2

−1 0 1

2 x 10

⁻¹⁴

Figure 5.2.7: Global error of the Q^{F M}₂ FMmethod with end points only and multiplicities all 2, for the equation y^′′(t) = −ωty(t)−cos(t), 0 ≤ t ≤ 100, with [1,0]^⊺ initial conditions, step-size h= ₁₀₀¹ and ω = 10⁶.

Chapter 6 Highly oscillatory non-linear systems

6.1 Waveform relaxation methods

In this chapter we present iteration techniques, namely, waveform relaxation algorithms for solving non-linear ordinary differential equations with associated initial conditions:

dtz(t) =f(t,z), z(0) =z₀, t∈[0, T], (6.1.1) where T ≥0 f : [0, T]×Rⁿ→Rⁿ, z₀ = [z1,0, z2,0, ..., zn,0]∈Rⁿ is the vector of initial values, andz(t) = [z1(t), z2(t), ..., zn(t)]∈Rⁿ is the solution vector.

In other words the system is written as follows, d

dtz1(t) =f1(t, z1, z2, ..., zn), z1(0) =z1,0

dtz2(t) =f2(t, z1, z2, ..., zn), z2(0) =z2,0

... ...

dtzn(t) =fn(t, z1, z2, ..., zn), zn(0) =zn,0.

6.1 Waveform relaxation methods

A simple example of waveform relaxation scheme, which maps the previous iterate z^(ν⁻¹⁾ into new iterate z^(ν) is,

dtz₁^(ν)(t) =f₁(t, z^(ν)₁ , z₂^(ν⁻¹⁾, z₃^(ν⁻¹⁾, ..., z_n^(ν⁻¹⁾), z₁^(ν)(0) =z_1,0 d

dtz₂^(ν)(t) =f2(t, z^(ν)₁ , z₂^(ν), z^(ν₃ ⁻¹⁾, ..., z_n^(ν⁻¹⁾), z₂^(ν)(0) =z2,0

... ...

dtz_n^(ν)(t) =fn(t, z₁^(ν⁾, z₂^(ν), z₃^(ν), ..., z_n^(ν⁾), z^(ν)_n (0) =zn,0. The above relaxation scheme is calledGauss-Seidelwaveform relaxation, similarly toGauss-Seideliterative method to solve linear and non-linear systems of algebraic equations. The iterative methods simplify the problem of solving large systems of differential equations of n variables to a sequence of differential equations of a single variable. Another example of iteration scheme is the Jacobi waveform relaxation method,

dtz₁^(ν⁾(t) =f1(t, z₁^(ν), z₂^(ν⁻¹⁾, z^(ν₃ ⁻¹⁾, ..., z_n^(ν⁻¹⁾), z₁^(ν)(0) =z1,0

dtz₂^(ν⁾(t) =f2(t, z₁^(ν⁻¹⁾, z₂^(ν), z^(ν₃ ⁻¹⁾, ..., z_n^(ν⁻¹⁾), z₂^(ν)(0) =z2,0

... ...

dtz_n^(ν⁾(t) =fn(t, z^(ν₁ ⁻¹⁾, z^(ν₂ ⁻¹⁾, z₃^(ν⁻¹⁾, ..., z_n^(ν)), z_n^(ν)(0) =zn,0.

In both cases the iterations scheme picks up initial conditions as an initial approximation z⁽⁰⁾(t)

z⁽⁰⁾_i (t) =zi,0, t ∈[0, T], i= 1, ..., n,

which is defined along the whole interval of integration. Easy to notice that unlikeJacobialgorithm, theGauss–Seidelmethod is sequential, i.e. one have to solve equations one after another, while the Jacobi scheme can be solved in parallel, simultaneously.

Let us definewaveform relaxation operator F as z^(ν)=F(z^(ν⁻¹⁾).

6.1 Waveform relaxation methods

Then a one-step iteration method can be written as d

dtz^(ν)(t) =F(t,z^(ν⁻¹⁾,z^(ν)), z^(ν)(0) =z₀, where F(u, v) is defined so, that F(t, v, v) =f(t, v).

The well known Picard iteration, as well as Jacobi and Gauss–Seidel schemes can be written in the formulae,

F_i(t, u, v) =F_i(t, u1, u2, ..., ui, ..., un), (Picard), F_i(t, u, v) =F_i(t, u1, u2, ..., ui−1, vi, ui+1, ..., un), (Jacobi), F_i(t, u, v) =F_i(t, v1, v2, ..., vi−1, vi, ui+1, ..., un), (Gauss-Seidel).

Convergence analysis ofwaveform relaxationmethods is presented [Van93], [Nev89a], [Nev89b] and [MN87]. The convergence of sequences {z^(ν)}^∞ν=0 is analysed in the context of Banach spaces.

We consider continuous vector-valued functions defined on [0, T], i.e.

C([0, T];Rⁿ), with maximum of the norm, defined as:

kzk^T = max

t∈[0,T]kz(t)k, with vector norm k · k inRⁿ.

Definition 6.1.1 Let (X,k · k^X) be a normed linear space and the associated norm. An operator U:X→Xis called a contraction if there exists a γ, with 0≤γ <1, such that for all x, y ∈X it is true that

kU(x)−U(y)k ≤γkx−yk^X.

It is shown in [Van93] that thewaveform relaxationis a contraction map- ping.

Theorem 6.1.1 [Van93] Let X be a Banach space and U a contraction.

Then there is a unique x^⋆ ∈ X, such that U(x^⋆) = x^⋆. Moreover, if x⁽⁰⁾ is any point in X, and we define the sequence {x^(ν)}^∞ν=0 by x⁽¹⁾ = U(x⁽¹⁾), then x^(ν) →x^⋆ as ν→ ∞.

In document Numerical methods for systems of highly oscillatory ordinary differential equations (página 54-107)