Backward error analysis for variational discretisations of PDEs

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 $