Model-Adaptive Simulation of Hierarchical Shallow Water Moment Equations in One Dimension

TL;DR

The paper tackles efficient simulation of shallow free-surface flows by allowing spatial and temporal variation in model complexity through the Shallow Water Moment Equations (SWME).It develops a model-adaptive framework that partitions the domain into subregions modeled with different orders $M_H$ and $M_L$, guided by residual-based error estimators derived from a hierarchical SWME decomposition, and couples interfaces with PBC or CIF schemes to preserve stability and, in the CIF case, conservation for the first $M_L+2$ equations.Key contributions include two boundary-interface coupling strategies (PBC and CIF), explicit model-coarsening and model-refinement criteria, and an algorithm that updates the local SWME order in time across the domain.Numerical experiments on a dam-break collision validate the approach, showing accurate results and substantial runtime speedups (up to 60%) compared with fixed high-order SWME, with CIF and PBC delivering similar performance and robustness.

Abstract

Shallow free surface flows are often characterized by both subdomains that require high modeling complexity and subdomains that can be sufficiently accurately modeled with low modeling complexity. Moreover, these subdomains may change in time as the water flows through the domain. This motivates the need for space and time adaptivity in the simulation of shallow free surface flows. In this paper, we develop the first adaptive simulations using the recently developed Shallow Water Moment Equations, which are an extension of the standard Shallow Water Equations that allow for vertically changing velocity profiles by including additional variables and equations. The model-specific modeling complexity of a shallow water moment model is determined by its order. The higher the order of the model, the more variables and equations are included in the model. Shallow water moment models are ideally suited for adaptivity because they are hierarchical such that low-order models and high-order models share the same structure. To enable adaptive simulations, we propose two approaches for the coupling of the varying-order shallow water moment equations at their boundary interfaces. The first approach dynamically updates padded state variables but cannot be written in conservative form, while the second approach uses fixed padded state variable values of zero and reduces to conservative form for conservative moment equations. The switching procedure between high-order models and low-order models is based on a new set of model error estimators, originating from a decomposition of the high-order models. Numerical results of the collision of a dam-break wave with a smooth wave yield accurate results, while achieving speedups up to 60 percent compared to a non-adaptive model with fixed modeling complexity.

Figures (7)

• Figure 1: Canonical domain decomposition example: the left lower-order subdomain $\Omega_{M_L}$ is modeled by the lower-order $\text{SWME}_{M_L}$ and the right higher-order subdomain $\Omega_{M_R}$ is modeled by the higher-order $\text{SWME}_{M_R}$, with $M_L<M_R$. Modified from PBC, Fig. 1.
• Figure 2: Padded Buffer Cell with added moments $\widetilde{\alpha}_{M_L\,+1},\ldots,\widetilde{\alpha}_{M_R}$ in cell $\mathcal{C}_L$, yielding the padded vector $\widetilde{w}_L$. The flux between cells $\mathcal{C}_L$ and $\mathcal{C}_R$ is computed using the padded vector $\widetilde{w}_L$ instead of $w_L$. Modified from PBC, Fig. 2.
• Figure 3: Piecewise path $\Phi_{\mathrm{pw}}(s;\cdot)$\ref{['eq:adaptive_path']} is composed of the two segments $\Phi_\mathrm{low}(s;\cdot)$ and $\Phi_\mathrm{high}(s;\cdot)$. The linear path $\Phi_{\text{lin}}(s;\cdot)$ is shown for comparison.
• Figure 4: Initial height profile of the test case displayed in Table \ref{['tab:setup-general']}.
• Figure 5: Collision of a dam break wave with a smooth wave with friction parameters $\lambda=1$ and $\nu=0.1$ (friction case 1) at time $t_{end}=5$. (a): Solid blue line is the simulated water height from the reference solution. The red crosses and the blue plus signs are the orders in each cell using the PBC coupling and the CIF coupling, respectively. (b)-(d): The lower-order model is the $\text{SWME}_1$ (solid red line). The higher-order model is the $\text{SWME}_5$ model (blue dash-dotted line). The A-SWME-PBC (black dashed line) and the A-SWME-CIF (green dotted line) yield similar results, close to the higher-order model, for the height $h$, mean velocity $u_m$, and first moment $\alpha_1$.

Theorems & Definitions (14)

• Remark 3.1
• Remark 3.2
• Definition 3.1: Model-coarsening criteria
• Definition 3.2: Model-refinement criteria
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Definition 3.3: Padded Buffer Cell (PBC) coupling
• Remark 3.6
• Remark 3.7