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.

翻译：我们提出了一种求解初边值问题的新方法。该方法基于作用量原理，其中坐标映射被纳入动力学自由度。这一具有重参数化不变性的作用量在抽象参数空间中构建，并与时空坐标相关的能量密度标度分离了坐标映射与传播场的动力学。将坐标视为从属（即动力学）变量，既可在离散化作用量的同时保持所有时空对称性，又为自动自适应网格细化（AMR）提供了基础。离散化后未破缺的时空对称性还确保了相应的连续诺特荷保持精确守恒。此外，坐标映射的存在为边界条件的选择提供了新自由度。我们利用最新发展的正则化求和分部有限差分算子，给出了$1+1$维波传播的显式数值示例。