Mapped Tent Pitching (MTP) Algorithm

• Mapped Tent Pitching (MTP) is a framework for solving hyperbolic PDEs using adaptive, tent-shaped spacetime domains that respect causality constraints.
• It maps physical tents to cylindrical reference domains, enabling high-order convergence and local explicit or implicit time-stepping across nonuniform meshes.
• The algorithm supports parallel computation and applies to various systems such as Maxwell, Euler, and acoustic wave equations with rigorous stability and error analysis.

The Mapped Tent Pitching (MTP) algorithm provides a spatially and temporally adaptive framework for the numerical solution of hyperbolic partial differential equations (PDEs), utilizing tent-shaped spacetime domains, causality-respecting unstructured meshing, and a mapping to separable space–time tent cylinders. The MTP approach enables local explicit or implicit time-stepping across spatially nonuniform meshes, achieving high-order convergence efficiently and supporting parallel computation. This class of methods has foundational support in the context of Friedrichs systems, linear symmetric hyperbolic systems, nonlinear conservation laws (Euler), Maxwell equations, and acoustic wave equations (Gopalakrishnan et al., 2016, Drake et al., 2021, Gopalakrishnan et al., 2019, Gopalakrishnan et al., 2015).

1. Spacetime Tent Mesh Construction and Causality Constraint

The tent pitching strategy operates on a bounded spatial domain $\Omega_0\subset\mathbb{R}^N$ discretized by a conforming, typically simplicial mesh $\mathcal{T}$ (Gopalakrishnan et al., 2016). A continuous, piecewise-linear "time-front" $\tau_{i-1}\in P_1(\mathcal{T})$ is maintained, indicating, at each vertex $v$, the local time up to which the solution is known. Advancing the time at one vertex by a "tent-pole height" $k$ produces an increment $\tau_i(x) = \tau_{i-1}(x) + k\,\eta_v(x)$, where $\eta_v$ is the nodal hat-function for $v$.

The spacetime tent for vertex $v$ is defined as $$K_i = \{\,(x,t):\,x\in\omega_v,\;\tau_{i-1}(x)<t<\tau_i(x)\,\}$$, with $\omega_v$ the set of elements adjoining $v$. The causality constraint governing the tent-pole height enforces, for every mesh edge $e=uv$,

$(\tau_i(u)-\tau_i(v)) / |u-v| \leq C_c / c_e,$

where $C_c$ is determined by mesh geometry and $c_e$ approximates the maximum wave speed on elements adjacent to $e$ (Gopalakrishnan et al., 2016, Gopalakrishnan et al., 2019, Drake et al., 2021). This ensures that the local solution can be progressed without violating the domain of dependence intrinsic to hyperbolic systems. Front advancement occurs in causally valid sets of non-intersecting vertex patches, permitting spacetime concurrency (Gopalakrishnan et al., 2015).

2. Mapping Tents to Reference Space–Time Cylinders

The inherent geometric skew in each physical tent is overcome by a mapping to a cylindrical reference tent $\widehat{K}_i = \omega_v \times (0,1)$, using:

$$\Phi(\hat{x},\hat{t}) = \big(x = \hat{x},\; t = (1-\hat{t})\tau_{i-1}(\hat{x}) + \hat{t}\,\tau_i(\hat{x})\big).$$

With definitions $$\delta(\hat{x}) = \tau_i(\hat{x}) - \tau_{i-1}(\hat{x})$$ and $$\phi(\hat{x}) = \tau_{i-1}(\hat{x}) + \frac{1}{2}\delta(\hat{x})$$, the Jacobian is block-triangular:

$$D\Phi = \begin{pmatrix} I_N & 0 \ \nabla_{\hat{x}}\phi & \delta \end{pmatrix} ,\quad \det D\Phi = \delta \neq 0.$$

The PDE transformation uses the Piola identity to obtain a mapped system on $\widehat{K}_i$, effecting separability of space and time variables (Gopalakrishnan et al., 2016, Drake et al., 2021, Gopalakrishnan et al., 2019). For generic symmetric systems $$\partial_t\,g(x, t, u) + \nabla_x\cdot f(x, t, u) + b(x, t, u) = 0$$,

$$\partial_{\hat{t}}[\,\hat{g}(\hat{x},\hat{t},\hat{u}) - \hat{f}(\hat{x},\hat{t},\hat{u})\phi\,] + \nabla_{\hat{x}}\cdot[\delta\,\hat{f}(\hat{x},\hat{t},\hat{u})] + \delta\,\hat{b}(\hat{x},\hat{t},\hat{u}) = 0.$$

This transformed conservation law is the basis for numerical discretization.

3. Spatial and Temporal Discretization Strategies

3.1 Spatial Discretization

MTP schemes utilize standard finite element (FE) or discontinuous Galerkin (DG) spaces on the spatial patch $\omega_v$ corresponding to each tent. Two strategies are prevalent:

• Discretization of $\hat{u}$: Seek $$\hat{u}_h(\hat{x},\hat{t}) = \sum_n u_n(\hat{t})\psi_n(\hat{x}),\; \psi_n\in X_i$$, yielding a time-dependent ODE system with mass-like matrices evolving with $\hat{t}$.
• Direct discretization of $U = \hat{g} - \hat{f}\cdot\phi$: Write $U_h = \sum_n U_n(\hat{t})\psi_n(\hat{x})$, producing ODEs with constant mass matrices (Gopalakrishnan et al., 2016).

Advective Friedrichs problems are cast in first-order systems with explicit, trace-conforming FE spaces engineered to satisfy weak continuity at boundary inflow–outflow junctions (Gopalakrishnan et al., 2015).

3.2 Time Integration

• Explicit methods: Standard explicit RK or multi-step methods can advance in $\hat{t}$, enforced by local CFL-type bounds, typically $\Delta\hat{t}\leq C_h h/p^2$ (with $h$ mesh size, $p$ polynomial degree). A salient feature is no global $\Delta t$ restriction; each tent’s temporal increment adapts locally to mesh resolution and wave speed (Gopalakrishnan et al., 2016).
• Locally implicit methods: Implicit RK (e.g., Radau IIA) schemes, obviating CFL restrictions, are feasible owing to the low-dimensional ODEs per tent.
• Structure-aware Taylor (SAT) methods: For Maxwell equations, explicit Taylor expansions utilize the affine structure of $M(\hat{t})$ to achieve high order and avoid the stage-order limitations of classical explicit RK (Gopalakrishnan et al., 2019). With $q$ substeps per tent, global order $O(h^{p+1})$ is attained for spatial degree $p$ (Gopalakrishnan et al., 2019, Drake et al., 2021).

4. Stability, Convergence, and Error Analysis

Stability is rigorously established at multiple levels:

• Exact tent-wise stability derives from the decrease of the norm $\|\hat{u}\|_{M(\hat{t})}$ along each tent, where $M(\hat{t})$ is the mapped mass matrix (Drake et al., 2021).
• Semidiscrete DG solutions propagate front-norm stability across spacetime layers, implying non-expansiveness global-in-time.
• Error analysis yields local semidiscrete errors $O(h^{p+1})$, and global errors $O(h^{p+1/2})$ (for sum over tents) when using polynomial degree $p$, and shows that temporal SAT order $s = p+1$ preserves spatial order (Drake et al., 2021, Gopalakrishnan et al., 2019).
• The causality constraint guarantees coercivity of $M(\hat{t})$, ensuring invertibility and compatibility with the hyperbolic nature of the PDE (Gopalakrishnan et al., 2019).

5. Exemplary Applications

MTP algorithms have demonstrated effectiveness on several prototype systems:

• Acoustic wave equations: Locally implicit MTP with mixed FE spaces achieves high order and stability, uncoupled from polynomial degree by CFL. Numerical tests confirm $O(h^p)$ $L^2$ error (Gopalakrishnan et al., 2016).
• Euler equations: Explicit MTP schemes with DG discretization and entropy viscosity provide closed-form updates per tent, with local CFL, necessitating additional viscosity time-step for stability in the presence of shocks (Gopalakrishnan et al., 2016).
• Maxwell equations: Explicit structure-aware Taylor methods validated high-order convergence and explicit propagation without order reduction, exploiting the affine-in-time mass matrix (Gopalakrishnan et al., 2019).
• One-dimensional Friedrichs systems: Low-order explicit updates form CTCS-like stencils with second-order convergence on uniform meshes, matching standard schemes in error decay and stability (Gopalakrishnan et al., 2015).

6. Algorithmic Implementation and Parallelization

Algorithmic steps are uniform across system types, comprising:

1. Initialization: Set $\tau\equiv0$ and assign initial solution values.
2. Tent pitching: Compute admissible tent-pole heights per vertex under causality.
3. Selection: Identify maximal independent sets of vertices for parallel tent advancement.
4. Local solves: For each tent, perform mapping, assemble local FE/DG matrices, and apply time stepping (explicit or implicit).
5. Solution propagation: Update solution on tent tops and advance time-fronts.
6. Synchronization and update: Prepare for downstream tent marches.

Data structures utilize adjacency lists, local front heights, tent-patch quadrature data, and sparse/matrix-free local operators. The number of tents to reach $T$ scales as $O(h^{-N})$, with per-tent complexity $O(N_{\text{dof}}^v\times p^N)$ (Drake et al., 2021). Concurrency is intrinsic to the algorithm: non-overlapping tents can be processed simultaneously, greatly benefiting from modern multicore and GPU architectures (Gopalakrishnan et al., 2016, Gopalakrishnan et al., 2019).

7. Key Properties, Limitations, and Outlook

Mapped Tent Pitching algorithms combine local time-adaptivity, high-order accuracy (spatial and temporal), causality assurance, and computational efficiency. The explicit algorithm is matrix-free and cache-friendly, while implicit variants offer unconditionally stable progression per tent (Gopalakrishnan et al., 2016). The separation of space and time in mapped tents facilitates reuse of mature spatial solvers and high concurrency.

A plausible implication is that the structure-aware Taylor time integration surmounts the classical explicit RK order reduction on variable-mass ODEs, thus unlocking explicit, high-order, fully local solvers for complex systems with highly unstructured meshes. However, local CFL-like constraints remain for explicit methods, particularly for nonlinear or strongly advective problems.

Mapped Tent Pitching’s unique combination of causality-respecting spacetime mesh generation and high-order separated discretization positions it as a robust framework for large-scale, adaptive simulation of hyperbolic systems. Subsequent developments have focused on rigorous error bounds, extended system classes (Friedrichs, Maxwell, Euler), and scalable implementations, with algorithmic refinements supporting both explicit and implicit local solvers (Gopalakrishnan et al., 2016, Drake et al., 2021, Gopalakrishnan et al., 2019, Gopalakrishnan et al., 2015).