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.
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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 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 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)) $
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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.
Dynamics of the amplitude variable
Phase portrait of the amplitude variables
Evaluation of the conserved quantity
Dynamics of the amplitude variable
Numerical integration of the ODE (27) truncated after
When
Interpretation of (4) as a multistep formula for