Stability

For each value of ξ, Eq. (5.12) admits four values ofω, two of which closely approximate the value ofω in the continuous dispersion relation for small values ofξ. A comparison of the discrete and continuous dispersion relations from Eqs. (5.12) and (5.6) respectively is shown in Figure 5.1.

Generally, such extra solutions to the discrete dispersion relation indicates that the numerical integrator supports spurious modes, which appear as unwanted parasitic waves in the numerical solution to the PDE. However, as will be shown below, for RK and PRK methods applied to multi-Hamiltonian PDEs that satisfy the conditions of Theorem 3.4.1, these extra solutions to the discrete dispersion relation have relevance and do not correspond to parasitic waves.

5.1.1 Stability

It was noted in Section 3.4.2 that for the class of multi-Hamiltonian PDEs satisfying the conditions of Theorem 3.4.1, the variablespcan be eliminated and the multi-Hamiltonian PDE can be written in the form

q_xx =f(q,q_t), (5.13) where q _∈ Rrank(K)_{. Now, if a periodic plane wave solution of the form of Eq. (5.2)} is assumed, then q_t = iωq and discretising Eq. (5.13) in space is equivalent to finding periodic solutions of the ODEs

−3 −2 −1 0 1 2 3 −4 −3 −2 −1 0 1 2 3 4 ξ∆x

ω∆x=±(ρ(∆x)2+11+cos(ξ∆x)±(73+70cos(ξ∆x)+cos2(ξ∆x))1/2)1/2

ω

∆

x

Figure 5.1: A comparison between the discrete dispersion relation (solid line) forρ(∆x)2 = 1 and the continuous dispersion relation (dashed line) forρ= 1.

Furthermore, iff(q) is linear then Eq. (5.14) can be decoupled into a set of harmonic oscillators with frequencies ωγ forγ = 1, . . . ,rank(K). This gives a dispersion relation,

−ξ2 =ωγ, (5.15)

for each harmonic oscillator subsystem labelled byγ, while the discrete dispersion relation for each harmonic oscillator subsystem is that of the PRK discretisation applied to a harmonic oscillator of frequency ωγ.

Suppose that such a PRK discretisation applied to a harmonic oscillator of frequency ωγ gives a linear map

qγ _7→R(∆xωγ)qγ. (5.16) If the coefficients of the RK or PRK method satisfies the symplecticity condition (Eq. (3.6)), then R has determinant 1. Furthermore, if the trace of R is at most 2 in absolute

value, then the map (5.16) is conjugate to a rotation by angle ξ where tr(R) = 2 cos(ξ). (R is known as the stability function and, for an RK method, takes the form [19]

R(z) = 1 +zbT(I₋zA)−11. (5.17) For Gaussian RK methods, it is an approximation to ez with an error of _O((∆x)2r+1).)

The stability of such a RK or PRK discretisation as a time integrator can be determined as the largest value of the time step ∆t such that _|tr(R)_{| ≤} 2, i.e., the largest value of ∆t such that the map (5.16) remains conjugate to a rotation. Similarly, the stability of such a RK or PRK discretisation in space can also be determined from tr(R). For a given wave number, ξ, in space, the values of ωγ _{such that tr(R(∆xω}γ_{)) = 2 cos(ξ) determine}

the dispersion relation. If the RK or PRK method has r stages at r distinct quadrature points and the modes of each harmonic oscillator subsystem are to be periodic, then there must be r distinct values of ω. If there are fewer than r distinct values of ω then the method will, in general, be unstable.

For an r-stage Gaussian RK discretisation in space there are preciselyrmodes for each value of ξ, thus, if the method is well defined, it will be stable. Furthermore, for these methods the stability function is invertible, in general, and _|tr(R)_{| ≤} 2 for all values of ωγ. Therefore, for this class of multi-Hamiltonian PDEs, the discrete dispersion relation is conjugate to the continuous dispersion relation. Moreover, it is monotonic and continuous. For an r-stage Lobatto IIIA–IIIB PRK discretisation in space there are preciselyr₋1 modes for each value of ξ (as is hinted at in Figure 5.1, the dispersion relation for the 3- stage method has twice as many modes as the continuous dispersion relation), however the last mode is not unstable as the last quadrature point coincides with the first quadrature point of the next cell and there is effectively onlyr₋1 active variables per cell. Thus the explicit ODEs formed by applying a Lobatto IIIA–IIIB discretisation in space to a multi- Hamiltonian PDE satisfying the conditions of Theorem 3.4.1 is a stable discretisation.

Furthermore, the regions where _|tr(R)_|>2 do not correspond to instability (as they would do in a time discretisation), but rather to jumps in the value of ω in the dispersion relation. Therefore, for this class of multi-Hamiltonian PDEs, the discrete dispersion relation is conjugate to a portion of the continuous dispersion relation. This conjugacy is monotonic but no longer necessarily continuous, as it was for Gaussian RK discretisations. A numerical demonstration of the stability of the Lobatto IIIA–IIIB discretisation in space is given in Figures 5.2, 5.3 and 5.4. To obtain these figures, the NLS equation is discretised in space with 3-stage Lobatto IIIA–IIIB and in time with 2-stage Lobatto IIIA–IIIB. The spatial domain is periodic on [0, L], where L= 4p(2)π, and divided into 256 equally spaced nodes. The initial conditions are given by Pi,1 = f(i∆x), Pi,2 = f((i+1₂)∆x) andQi,1 =Qi,2 = 0, wheref(x) = 1.5(1−0.1 cos(2πx/L)). This integrator is then stepped forwards in time for 107 _{steps with a step size of 10}−4_{. (This step size}

0 5 10 15 0 100 200 300 400 500 600 700 800 900 1000 5 10 15 20 Norm (p2+q2) 0 200 400 600 800 1000 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 Energy error

Figure 5.2: A waterfall plot of the norm (p2 +q2) and the energy error for the NLS equation.

is chosen such that the quadratic equations in Eq. (4.12) can be solved. For many discretisations in time, the step size will also be restricted by ∆t < C(∆x)2 in order for the integrator to remain stable.)

In Figure 5.2, a waterfall plot of the norm (p2+q2) is shown on the left, while a plot of the error in the total energy is shown on the right. The energy behaviour is what one would expect from a symplectic integrator, that is, the total energy is approximately conserved by the integrator. Remarkably, for symmetric initial conditions this integrator appears to preserve the symmetry of the solution exactly. In figure 5.3, the solution (p and q) at t = 1000 (i.e., after 107 steps) is given. This solution appears to be smooth, i.e., there appears to be no high frequency oscillations in space, which is an indication of the stability of the integrator. The Fourier transform of p at t = 1000 in Figure 5.4 (the Fourier transform of q is almost identical) confirms the lack of any high frequency oscillations in space.