Strongly stable dual-pairing summation by parts finite difference schemes for the vector invariant nonlinear shallow water equations -- I: Numerical scheme and validation on the plane

Abstract

We present an energy/entropy stable and high order accurate finite difference (FD) method for solving the nonlinear (rotating) shallow water equations (SWEs) in vector invariant form using the newly developed dual-pairing and dispersion-relation preserving summation by parts (SBP) FD operators. We derive new well-posed boundary conditions (BCs) for the SWE in one space dimension, formulated in terms of fluxes and applicable to linear and nonlinear SWEs. For the nonlinear vector invariant SWE in the subcritical regime, where energy is an entropy functional, we find that energy/entropy stability ensures the boundedness of numerical solution but does not guarantee convergence. Adequate amount of numerical dissipation is necessary to control high frequency errors which could negatively impact accuracy in the numerical simulations. Using the dual-pairing SBP framework, we derive high order accurate and nonlinear hyper-viscosity operator which dissipates entropy and enstrophy. The hyper-viscosity operator effectively minimises oscillations from shocks and discontinuities, and eliminates high frequency grid-scale errors. The numerical method is most suitable for the simulations of subcritical flows typically observed in atmospheric and geostrophic flow problems. We prove both nonlinear and local linear stability results, as well as a priori error estimates for the semi-discrete approximations of both linear and nonlinear SWEs. Convergence, accuracy, and well-balanced properties are verified via the method of manufactured solutions and canonical test problems such as the dam break and lake at rest. Numerical simulations in two-dimensions are presented which include the rotating and merging vortex problem and barotropic shear instability, with fully developed turbulence.

Figures (15)

• Figure 1: Smooth boxcar function used for the hyper-viscosity operator.
• Figure 2: Numerical eigenspectra of the DP operators of order 6 with $N = 501$, $g=H = 1$, for the three main BCs considered using SAT terms: mass flux, velocity flux and transmissive BCs. The first two are energy conserving without hyper-viscosity and maintains zero real parts up to machine error, while adding hyper-viscosity and utilising transmissive BCs should dissipate energy, which yields non-positive real parts. All spectra clearly indicate strict stability of the eigensystem.
• Figure 3: Numerical eigenspectra of the linearised non-linear operator for the DP operators of order 6 with $N = 501$, $g=1$, with the three main BCs considered using SAT terms: mass flux, velocity flux and transmissive BCs. The first two are energy conserving without hyper-viscosity and maintains zero real parts up to machine error, while adding hyper-viscosity and utilising transmissive BCs should dissipate energy, which yields non-positive real parts. All spectra clearly indicate strict stability of the eigensystem.
• Figure 4: Plot of the height for the canonical lake at rest problem at end time $t = 5$, where the bathymetry $b(x)$ is defined via \ref{['eqn:b(x)']}, which is non-smooth about $x \in (8,12)$. The simulation is initialised with $h + b = 0.5$, $N = 201$, $L = 25$. Our numerical scheme maintains the initial condition up to machine precision.
• Figure 5: Time evolution of the lake at rest profile with 10% perturbations on $h$ with transmissive BC. The waves transmit out the right and left boundaries to maintain an overall steady state flowfield similar to the canonical lake at rest problem.

Theorems & Definitions (37)

• Definition 2.1
• Definition 2.2
• Lemma 1
• proof
• Definition 2.3
• Theorem 2
• proof
• Remark 1
• Lemma 3
• proof