Technical Summary: Mimetic Spectral Element Advection

Problem Statement

The paper addresses the numerical discretization of the linear advection equation for differential forms on bounded domains. The classical scalar advection problem, $\partial_t \rho + \nabla \cdot (v\rho) = 0$, is generalized within a differential geometric framework as:
$$\frac{\partial \alpha^{(k)}}{\partial t} + \mathcal{L}_v \alpha^{(k)} = 0$$
where $\alpha^{(k)}$ is a $k$-differential form (representing scalars, vectors, or higher-dimensional quantities depending on $k$), $v$ is a prescribed Lipschitz continuous velocity field, and $\mathcal{L}_v$ is the Lie derivative. The challenge lies in developing a discretization that preserves the underlying geometric structures of the physical laws, ensures local mass conservation, and achieves spectral accuracy while distinguishing between quantities evaluated at specific time instants and those integrated over time intervals.

Methodology

The proposed method, Mimetic Spectral Element Advection, extends the mimetic framework established in prior work [4] by incorporating the Lie derivative via Cartan's homotopy formula. The approach is built upon three core pillars:

1. Differential Geometric Foundation
The method utilizes the language of differential forms. The Lie derivative $\mathcal{L}_v$ is expressed using Cartan's homotopy formula:
$$\mathcal{L}_v \alpha^{(k)} = d\iota_v \alpha^{(k)} + \iota_v d\alpha^{(k)}$$
where $d$ is the exterior derivative and $\iota_v$ is the interior product. A crucial aspect of the formulation is the adjoint relationship between the interior product and the wedge product, defined via the $L^2$ inner product:
$$(\iota_v \alpha^{(k)}, \beta^{(k-1)})_{L^2} = (\alpha^{(k)}, v^\flat \wedge \beta^{(k-1)})_{L^2}$$
This duality allows physical quantities represented by interior products to be mapped to their dual differential 1-forms.

2. Spatial Discretization (Mimetic Spectral Elements)
The spatial domain is discretized using a cell complex consisting of points, line segments, surfaces, and volumes. The space of smooth $k$-forms is approximated by a finite-dimensional space $\Lambda_h^k$ spanned by basis forms $\epsilon_i^{(k)}$.

• Basis Construction: The basis functions are constructed using piecewise polynomial expansions (tensor products) on quadrilateral elements. In 1D, 0-forms are represented by Lagrange polynomials evaluated at Gauss-Lobatto nodes, while 1-forms are represented by "edge polynomials" derived to satisfy specific integral properties.
• Commuting Projection: A projection operator $\pi_h$ is defined such that it commutes with the exterior derivative ($\pi_h d = d \pi_h$). This ensures that the discrete exterior derivative $d_h$ is represented by incidence matrices containing only values $\{0, 1, -1\}$, preserving the topological structure of the continuous operators.

3. Temporal Discretization (Canonical Mimetic Integrator)
The time integration employs an arbitrary-order symplectic operator derived from canonical Gauss collocation integrators [5].

• Staggered Time Nodes: The method distinguishes between two sets of time nodes:
  • Gauss-Lobatto nodes ($t_k$): Where the solution variables (e.g., $\rho$) are defined.
  • Gauss nodes ($\tilde{t}_q$): Where the time derivatives (fluxes) are evaluated.
• Discrete Evolution: This staggering results in a discrete integrator where the change in the solution over an interval is equated to the flux evaluated at the internal Gauss nodes. This structure mirrors leap-frog methods and the implicit midpoint rule, ensuring symplectic properties.

4. Discrete Interior Product
The interior product $\iota_v$ is discretized by enforcing the duality pairing (Eq. 16) in the discrete setting. This leads to a system where the discrete fluxes are computed by solving a linear system involving the inner products of the basis forms and the velocity field.

Key Contributions

• Geometric Consistency: The scheme explicitly incorporates the Lie derivative using Cartan's formula, ensuring the discretization respects the metric-free representation of differential operators and their Hilbert adjoints.
• Spectral Accuracy: The method utilizes high-order polynomial basis functions, enabling spectral convergence in space.
• Local Mass Conservation: The use of incidence matrices and the specific structure of the discrete exterior derivative guarantees local mass conservation.
• Time-Stepping Distinction: The framework rigorously separates quantities evaluated at time instants from those integrated over intervals, utilizing a staggered time grid to achieve high-order symplectic integration.

Numerical Results

The authors present numerical experiments on 2D domains with periodic boundary conditions, testing the advection of sine waves and sine bells in constant and Rudman vortex velocity fields.

• Convergence: The method demonstrates algebraic $h$-convergence of order $(p+1)$ and spectral $p$-convergence, provided the time integration error does not dominate the spatial error.
• Time Integration Accuracy: The error in the solution is shown to be dependent on the order of the time integration scheme ($p_t$). If the time scheme is sufficiently accurate, the initial discretization error is preserved; otherwise, artificial dispersion increases over time.
• Mass Conservation: The total mass error remains at machine zero for the first $10^3$ time steps and stays below $10^{-12}$ even after $2 \times 10^4$ steps, demonstrating excellent conservation properties.
• Reversibility: A test involving the advection of a sine wave in a Rudman vortex, followed by a reversal of the flow direction, shows that the method can recover the initial solution, proving the reversibility of the integration method.
• Artificial Dispersion: The paper notes that artificial dispersion is a function of the time integration order; lower-order time schemes introduce dispersion errors that depend on the frequency of the advected wave.

Significance and Claims

The paper claims that the derived scheme successfully combines spectral accuracy with local mass conservation within a physics-compatible (mimetic) framework. By extending the mimetic framework to include the Lie derivative via Cartan's homotopy formula, the authors provide a discretization that clarifies the geometric structure of the advection equation. The method is presented as a robust approach for solving advection problems where preserving the underlying geometric and topological properties of the physical laws is essential. The authors modestly note that while the spatial discretization is highly accurate, the overall accuracy and dispersion characteristics are contingent upon the order of the time integration scheme used.