Exact symmetry conservation and automatic mesh refinement in discrete initial boundary value problems

TL;DR

This work tackles the problem of discretizing initial boundary value problems without breaking space-time symmetries or conserving Noether charges. It introduces a reparameterization-invariant action defined on abstract parameters $(\tau,\vec{\sigma})$ with dynamical coordinate maps $t(\tau,\vec{\sigma})$ and $\vec{x}(\tau,\vec{\sigma})$, coupled to fields via a scale $T$; discretization occurs on the abstract parameter space, preserving continuum symmetries and yielding exact Noether charges. A Schwinger-Keldysh/Galley-like doubling is used to formulate causal IBVPs, with Lagrange multipliers enforcing initial, boundary, and connecting conditions, and summation-by-parts (SBP) operators providing a stable, symmetry-preserving discretization. The proof-of-principle in 1+1 dimensions demonstrates exact energy conservation via a discrete Noether charge, and automatic adaptive mesh refinement (AMR) emerges from the dynamical time mapping, while the numerical results show convergence to the continuum solution and robust boundary handling. The approach promises improved symmetry preservation, stability, and boundary-flexibility for IBVPs, with clear paths toward higher dimensions and gauge-field extensions.

Abstract

We present a novel solution procedure for initial boundary value problems. The procedure is based on an action principle, in which coordinate maps are included as dynamical degrees of freedom. This reparametrization invariant action is formulated in an abstract parameter space and an energy density scale associated with the space-time coordinates separates the dynamics of the coordinate maps and of the propagating fields. Treating coordinates as dependent, i.e. dynamical quantities, offers the opportunity to discretize the action while retaining all space-time symmetries and also provides the basis for automatic adaptive mesh refinement (AMR). The presence of unbroken space-time symmetries after discretization also ensures that the associated continuum Noether charges remain exactly conserved. The presence of coordinate maps in addition provides new freedom in the choice of boundary conditions. An explicit numerical example for wave propagation in $1+1$ dimensions is provided, using recently developed regularized summation-by-parts finite difference operators.

Figures (12)

• Figure 1: (Top) Sketch of the conventional approach to IBVPs: space-time coordinates are designated as independent variables and the field $\phi(t,x)$ propagates on the background of this space-time scaffold. When discretizing space-time as a hypercubic grid, the field $\phi(t_i,x_j)$ is resolved in a similarly regular fashion. Since discretized space-time is unable to accommodate infinitesimal symmetry transformations, the continuum Noether charge is not preserved. (Bottom) Sketch of our novel approach to IBVPs: A set of abstract parameters $(\tau,\sigma)$ is designated as independent parameters. Both the field $\phi(\tau,\sigma)$ as well as dynamical coordinate maps $t(\tau,\sigma)$ and $x(\tau,\sigma)$ evolve on the background of the $(\tau,\sigma)$ parameters. The evolution of the coordinate maps can be highly non-linear depending on the field dynamics. Discretizing the $(\tau,\sigma)$ parameters leaves the values of the coordinate maps continuous. In turn when expressing the physical field solution in terms of space-time coordinates $\phi(t(\tau_i,\sigma_j),x(\tau_i,\sigma_j))$, one finds in general that a non equidistant space-time grid emerges, which automatically adapts in resolution to the dynamics of the field. Since the discrete action retains its continuum symmetries the continuum Noether charge remains exactly conserved.
• Figure 2: (left) The conventional BVP formulation of classical field theory, where both initial data $\phi(\tau^{\rm i},\vec{\sigma})$ and final data $\phi(\tau^{\rm f},\vec{\sigma})$ (in green) are provided a priori. While different field configurations exist that connect these two datasets (gray sheets), there exists a unique field configuration $\phi_{\rm cl}$ (in red) that constitutes the critical point of the action. It is this configuration that represents the configuration realized in nature. (right) The doubled d.o.f. construction of galley_classical_2013, necessary to formulate causal IBVPs on the action level. Provided initial data on values $\phi(\tau^{\rm i},\vec{\sigma})$ and derivatives $\dot \phi(\tau^{\rm i},\vec{\sigma})$ (in green), one constructs a double shooting method. One copy of the field evolves forward (arrows to the right) and one copy evolves backward from the final state of the forward branch (arrows to the left). While different field configuration pairs exists that accommodate the initial data on the forward branch (e.g. gray sheets), only a single one (red and blue sheets) fulfills the requirement that it constitutes the critical point of the action with doubled degrees of freedom and that the forward and backward branch solutions agree (for more details see e.g. Rothkopf:2022zfb).
• Figure 3: Initial and boundary values provided for the dynamic field $\phi$ as red points and initial values of the time mapping $t$.
• Figure 4: (left) Eigenvalue spectrum of the unregularized SBP121 finite difference operators $\mathds{D}_\tau$ (red circles) and $\mathds{D}_\sigma$ (blue crosses) on a grid with $N_\sigma=24$ and $N_\tau=16$. Since $\Delta \tau<\Delta \sigma$ the purely imaginary eigenvalues of $\mathds{D}_\tau$ spread over a larger interval than for the spatial derivative. Each operator has exactly two zero modes depicted at the origin. (right) Eigenvalue spectrum of the regularized SBP121 operators (using the initial and boundary data from \ref{['fig:IBconfPhit']}) in affine coordinates. The zero modes of the SBP operator are lifted and the original physical zero mode, i.e. the constant function, is now associated with the unit eigenvalue.
• Figure 5: (top) Two Classical solution for the field $\phi$ (top) and the time-mapping $t$ (bottom) from the critical point of \ref{['eq:discrEsim']} evaluated on a grid with $N_\sigma=48$ and $N_\tau=60$ points. Note that the initial and boundary conditions of \ref{['fig:IBconfPhit']} are exactly obeyed.