Article Contents
Article Contents

# Backward error analysis for variational discretisations of PDEs

• *Corresponding author: Christian Offen
• In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby 'modified' equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.

Mathematics Subject Classification: Primary: 65D30, 35A15, 35B06, 35C07, 37K58; Secondary: 70H25, 70H50.

 Citation:

• Figure 1.  Illustration of Theorem 1.1. The left hand column gives the actions of a PDE and an associated ODE that governs its symmetric solutions such as travelling waves. The right hand column gives three Lagrangians of modified equations of a variational discretization. Top: of the discretization, containing arbitrarily high derivatives; middle: of its symmetric solutions, containing arbitrarily high derivatives; and bottom: of its symmetric solutions, containing first derivatives only. $\tilde{\mathcal L}$ can be regarded as a modified Lagrangian of $L^0$. ${\mathcal L}_\Delta$, $\mathcal L$ and $\tilde {\mathcal L}$ are formal power series in the step sizes

Figure 2.  Dynamics of the amplitude variable $\phi_1(\xi)$ for $\alpha \in \{0, 0.3, 0.5, 0.7\}$ for $V(a) = -\exp(-(a-1)^2)$ and the wave speed $c = 0.5$. (Initial condition $\phi_1(0) = \phi_2(0) = \dot\phi_1(0) = \dot\phi_2(0) = 0.1$)

Figure 3.  Phase portrait of the amplitude variables $\phi_1(\xi)$, $\phi_2(\xi)$ for $\alpha \in \{0, 0.1, 0.6\}$, $V(a) = -\exp(-(a-1)^2)$, the wave speed $c = 0.5$ and $\xi \in [-5, 10]$. (Initial condition $\phi_1(0) = \phi_2(0) = \dot\phi_1(0) = \dot\phi_2(0) = 0.1$)

Figure 4.  Evaluation of the conserved quantity $I_{\mathrm{rot}}$ (see 12) along a numerically computed trajectory shows round-off errors only (vertical axis is scaled by $10^{-14}$). Here $V(a) = -\frac 12 a -a^2$, $\alpha = -1$, $c = 2$. The integrator is the symplectic midpoint rule. Implicit equations are solved using fixed point iterations

Figure 5.  Dynamics of the amplitude variable $\phi_1(\xi)$ for $\alpha = 0$, $V(a) = -\exp(-(a-1)^2)$, $c = 0.5$ and $\Delta x \in \{0, 0.6, 1, 1.2\}$ for the modified equation truncated after $\mathcal O(h^3)$ terms.

Figure 6.  Numerical integration of the ODE (27) truncated after $\mathcal O(h^2)$ terms with $V(s) = -0.1 s^4 +s$, $\Delta x = 0.1$, $\Delta t = 0.15$, $\alpha = 0.3$, $c = 2$. All numerical computations have been performed in the Darboux variables $(\mathfrak q, \mathfrak p)$ of the continuous system using the implicit midpoint rule combined with fixed-point iterations. Therefore, the integration is symplectic modulo second order terms. The plots show a phase plot of a motion initialised at $(\mathfrak q, \mathfrak p) = (-0.11, -0.01, -0.1, 0.1)$ and the behaviour of the Hamiltonian $H$ of the exact system and the Hamiltonian $\mathcal H$ of the modified system truncated after $\mathcal O(h^2)$ terms as well as the behaviour of the conserved quantity of the exact system $I_{\mathrm{rot}}$ and of the modified system $I_{\mathrm{rot}}^{\mathrm{mod}}$ truncated after $\mathcal O(h^2)$ terms along the motion

Figure 7.  When $c \Delta t/\Delta x$ is rational, the functional equation (25) yields a multistep formula. The series parameter $h$ is set to 1. We use $V(s) = s^2$, $\Delta t = 0.15$, $\Delta x = 2 c \Delta t$. Let $\Delta \tau = c \Delta t$. To initialise the scheme, values at $\xi = \Delta \tau, 2\Delta \tau, 3\Delta \tau$ are obtained by integrating (27) truncated to 4th order with high accuracy with the initial condition $(\phi(0), \dot{\phi}(0)) = ((0.1, -0.05), (0, 0.1))$

Figure 8.  Interpretation of (4) as a multistep formula for $\frac mn = \frac{\Delta x}{c\Delta t}<1$. The variable $\hat \xi$ corresponds to $\xi - c \Delta t$ when comparing with (4) and $\Delta s = 2 c \Delta t$

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