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Semi-discrete Active Flux method for two-dimensional hyperbolic systems with Coriolis source terms

Nikhil Manoj, Wasilij Barsukow, Christian Klingenberg

arXiv:2609.27555v1math.NAcs.NA

TL;DR

The paper asks whether semi-discrete Active Flux can preserve non-trivial geostrophic stationary states in systems with Coriolis source terms. Fourier analysis establishes a discrete equilibrium for the linear system, while numerical experiments examine both linear and nonlinear cases.

  • Problem

    The linear shallow-water-related systems have non-trivial geostrophic stationary states, so numerical discretizations need to preserve these equilibria to avoid polluting perturbation waves with spurious waves.

    Source: §1 · p. 2

  • Method

    The paper analyzes semi-discrete Active Flux for linear and nonlinear systems with Coriolis source terms, using discrete Fourier analysis of the linear method’s evolution matrix.

    Source: §1 · p. 2

  • Results

    For the linear system, the method has a discrete stationary state corresponding to geostrophic equilibrium and preserves it without additional modifications; experiments validate this result.

    Source: §1 · p. 2, §6 · p. 18

  • Takeaways & Limitations

    The results indicate that Active Flux preserves linear geostrophic stationary states and performs well for nonlinear shallow-water equations, especially near geostrophic equilibrium.

    Source: §6 · p. 18

    The low-Froude, low-Rossby asymptotic geostrophic state is not necessarily a stationary solution of the nonlinear system.

    Source: §2.1 · p. 3

Abstract

from arXiv · show

We investigate the semi-discrete Active Flux numerical method on Cartesian grids applied to two-dimensional linear and nonlinear shallow water equations with Coriolis source terms. These hyperbolic systems admit non-trivial stationary solutions governed by a geostrophic equilibrium. In this setting, we analyze the stationarity-preserving properties of our semi-discrete Active Flux formulation. To do so for the linear system, we apply a discrete spatial Fourier transform to the semi-discrete method and analyze the resulting evolution matrix. We show that its discrete kernel is a non-trivial discretization of the geostrophic equilibrium, which implies that the method preserves a discrete geostrophic equilibrium state. This is an inherent feature of Active Flux and is achieved without any additional modifications. Numerical experiments confirm the method's ability to maintain geostrophic stationary states for the linear system. Additionally, we numerically investigate the performance of the Active Flux method for the nonlinear shallow water system.

1 Introduction

The paper studies whether semi-discrete Active Flux preserves geostrophic stationary states in shallow-water-related systems with Coriolis forces. It establishes a discrete equilibrium for the linear system and investigates performance for the nonlinear system.

1 Introduction

  • Well balancing preserves non-trivial stationary states on coarse meshes, important because physically relevant waves perturb these states and may otherwise be polluted by spurious waves.The paper frames this property as crucial for practical, long-time simulations.Source: §1 · p. 2
  • Semi-discrete Active Flux evolves cell averages conservatively and independently evolves interface point values using a method-of-lines spatial discretization with SSP-RK time integration.Its globally continuous spatial reconstruction avoids exact or approximate Riemann solvers.Source: §1 · p. 2
  • Earlier Active Flux studies found intrinsic stationarity preservation in some settings, while finite-volume and finite-difference methods generally require modifications to achieve it.Prior work also considered source-term problems including linear systems, shallow water with topography, and ideal MHD.Source: §1 · p. 2
  • The semi-discrete Active Flux method admits a discrete geostrophic equilibrium for the linear system, with stability and nonlinear-system performance also investigated.The rotational source term imposes the same stability threshold found for pure acoustics; nonlinear performance is reported especially in low Froude regimes.Source: §1 · p. 2

2 Hyperbolic systems with Coriolis source terms

This section develops the shallow-water setting and its geostrophic equilibria, then linearizes around rest to characterize stationary states of the resulting Coriolis-modified acoustics system.

2 Hyperbolic systems with Coriolis source terms

  • The paper considers dimensionless shallow-water equations over flat topography with Coriolis source terms, defining water depth, velocity, and the associated physical scales.The scaling uses Froude, Rossby, and Strouhal numbers, with gravitational acceleration and Earth’s angular velocity among the physical parameters.Source: §2 · p. 3

2.1 Low Froude - low Rossby asymptotics

  • In the low-Froude, low-Rossby regime with O(1) Strouhal number, the asymptotic analysis yields geostrophic equilibria with divergence-free leading-order velocity, but these need not be stationary solutions of the nonlinear system.The stated regime takes Fr = Ro = ϵ as ϵ tends to zero.Source: §2.1 · p. 3

2.2 Linearization around a state of rest

  • Linearizing around rest with constant height and zero velocity gives the standard linear acoustics equations augmented by a Coriolis source term.The linearization neglects O(δ^2) terms and defines p := η/Fr.Source: §2.2 · p. 4

2.3 Stationary solutions for the linear system

  • For the linear system, a Fourier mode is stationary when its amplitude lies in the kernel of iJ·k − S; the resulting kernel describes the geostrophic equilibrium.The kernel is spanned by (−ik_y a*, ik_x a*, c)^T, and the stationary relations are Fourier transforms of the geostrophic conditions.Source: §2.3 · pp. 4–5

3 The semi-discrete Active Flux method

The semi-discrete Active Flux method evolves cell averages and interface point values on a Cartesian grid using a globally continuous reconstruction. Its cell-average and point-value updates form an ODE system that is advanced with an SSP Runge–Kutta method.

3.1 Discretization and degrees of freedom

  • Each cell has nine local values—one average, four nodes, and four edge centers—but sharing leaves four effective cell degrees of freedom for a unique biparabolic reconstruction.Source: §3.1 · p. 5
  • The reconstruction is globally continuous across cells, although its spatial derivative traces need not coincide at cell interfaces.Source: §3.1 · p. 5

3.2 Semi-discrete numerical method

  • The semi-discrete method derives ODE updates for cell averages from the conservative formulation and point values from the quasilinear formulation, then advances them with an SSP Runge–Kutta integrator.Source: §3.2 · p. 5

3.2.1 Evolution of cell-averages

  • Cell-average evolution uses conservative fluxes approximated by Simpson’s quadrature; source-term averages are computed from cell averages, and continuity removes the need for a Riemann solver.Source: §3.2.1 · p. 6

3.3 Evolution of point values

  • Point-value updates use split Jacobians acting on left and right derivative traces for upwinding; omitting DP gives central Active Flux, while the alternative Rusanov splitting is not stationarity preserving.Source: §3.3 · pp. 6–7

3.4 Discretization in time

  • CFL ≤ 0.27 is required for the fully discrete method to remain stable in two spatial dimensions.The bound comes from the analysis of the semi-discrete method.Source: §3.4 · p. 7

4 Stationarity preservation

Section 4 analyzes stationarity preservation, dispersion, and fully discrete stability of semi-discrete Active Flux using Fourier methods. The analysis identifies a discrete stationary state for every wave number and reports fourth-order physical-mode errors and a tested stability threshold.

4.1 Stationarity preservation of semi-discrete Active Flux using the discrete Fourier transform

  • For every wave number, the semi-discrete evolution matrix has a one-dimensional kernel, yielding a discrete stationary state for linear acoustics with Coriolis sources.A discrete spatial Fourier ansatz reduces the method to a 12×12 complex evolution matrix whose kernel encodes stationary data.Source: §4.1 · pp. 8–9
  • The central Active Flux method has the same kernel as the upwind variant, and point-value numerical diffusion vanishes on that kernel.Source: §4.1 · p. 9

4.2 Dispersion analysis

  • Physical modes have fourth-order convergence in both dispersion and dissipation error, while nonphysical modes receive high numerical diffusion and are damped.Source: §4.2 · p. 10, Figure 2 · p. 10

4.3 Fully discrete stability analysis

  • With Δx = Δy = 1 and c = 1, time steps up to 0.275 remain stable for all wave numbers, while Δt ≥ 0.29 violates the stability threshold.Source: §4.3 · p. 11, Figure 3 · p. 11

5 Numerical experiments

The experiments test semi-discrete Active Flux on linear and nonlinear shallow-water systems with Coriolis terms. Linear tests show long-time stationarity behavior and third-order convergence, while nonlinear tests examine vortices and geostrophic adjustment.

5.1 Linear system

  • The linear stationary vortex remains well resolved to t = 10^3 with semi-discrete Active Flux, while the Rusanov-splitting solution degrades by t = 100.The Active Flux run uses a 20 × 20 grid; the Rusanov comparison uses a 40 × 40 grid.Source: §5.1 · p. 12, Figure 4 · p. 13, Figure 5 · p. 14
  • The numerical solution approaches the discrete steady state during an initial layer, whereas residuals for a different geostrophic discretization do not decay.The well-prepared discrete state is characterized by the Active Flux residuals.Source: §5.1 · p. 12
  • The linear-system method attains the desired third-order convergence.The experiment evolves the stationary-vortex initial datum to t = 1.0 across mesh sizes.Source: §5.1 · p. 12
  • For a well-prepared initial datum, L1 errors in u, v, and p remain close to machine precision through t = 10^3, indicating only round-off error.The plotted errors use the well-prepared datum as the reference.Source: §5.1 · p. 12, Figure 6 · p. 14
  • A localized pressure perturbation propagates without spurious waves under Active Flux, unlike with Rusanov splitting.The initial velocity is unperturbed; the comparison is shown at t = 0.3.Source: §5.1 · p. 14, Figure 7 · p. 15

5.2 Nonlinear shallow water equations

  • As the Froude parameter ε tends to zero, the relative water-depth error decays; with ε = 0.001, Rusanov splitting breaks down the stationary vortex by t = 500.The error is evaluated at t = 200 for the nonlinear stationary-vortex test.Source: §5.2 · p. 16, Figure 9 · p. 17, Figure 10 · p. 17
  • For the translated vortex at ε = 0.01, Active Flux preserves the vortex structure and amplitude through t = 20.The test uses a 40 × 40 grid with periodic boundaries in y and wall boundaries in x.Source: §5.2 · pp. 16–17
  • In the geostrophic-adjustment test, Active Flux approximates outward-propagating discontinuities and preserves the overall solution structure.The initial state is not in geostrophic equilibrium; the displayed water-height solutions are at t = 4 and t = 8.Source: §5.2 · p. 17

6 Conclusion

The paper analyzes semi-discrete Active Flux for linearized and nonlinear shallow-water equations with Coriolis source terms. Fourier analysis establishes discrete stationary states for the linearized system, and numerical experiments support the linear result and show good nonlinear performance near geostrophic equilibrium.

6 Conclusion

  • Fourier analysis establishes that the linearized system admits discrete stationary states corresponding to analytical geostrophic equilibrium, without additional modifications.Numerical experiments validate the linear-system analysis; nonlinear results are numerical, especially near geostrophic equilibrium.Source: §6 · pp. 17–18
  • The authors identify energy stability and extending the theoretical well-balanced analysis to the nonlinear system as future research.Source: §6 · p. 18
  • Numerical experiments validate the linear-system results and indicate good performance for nonlinear shallow-water equations, especially near geostrophic equilibrium.Source: §6 · p. 18

A Discrete differential operators in the Active Flux point value update

The appendix specifies where the discrete spatial derivatives used in the Active Flux point-value update are evaluated. It lists locations for both coordinate directions and notes that horizontal-edge derivatives are computed similarly.

A Discrete differential operators in the Active Flux point value update

B Update equations upon applying the Fourier transform

The appendix lays out the Fourier-transformed semi-discrete update equations by degree of freedom. It lists updates for nodal, vertical-edge, and horizontal-edge point values, as well as the cell average.

B Update equations upon applying the Fourier transform

C.1 Evolution submatrices for the Active Flux method

The evolution-matrix submatrices are derived from the Active Flux update formulas. Source-term effects are confined to diagonal submatrices because each update uses matching cell-average or point-value data.

C.1 Evolution submatrices for the Active Flux method

  • The evolution-matrix submatrices in (4.4) are derived using the update formulas in Section B.The supplied expressions include the submatrices of E.Source: §C.1 · p. 23
  • Source terms influence only the diagonal submatrices EAA, EEHEH, EEV EV, and ENN.The stated restriction follows from the source discretization using cell averages in the cell-average update and the respective point value in each point-value update.Source: §C.1 · p. 24
  • The source-term discretization uses cell averages for cell-average evolution and the respective point values for point-value evolution.This dependency explains why the source-term influence remains on the diagonal of E.Source: §C.1 · p. 24

C.2 Evolution submatrices for the central Active Flux method

The central Active Flux method yields an ODE system and an evolution matrix with submatrices specified in (4.4), following the procedure used for the central method in Section 4.

C.2 Evolution submatrices for the central Active Flux method

  • Following the procedure in Section 4, the central Active Flux method yields an ODE system of the form (4.3).Its evolution matrix has the form (4.4), with submatrices given there.Source: §C.2 · p. 24
  • The supplied submatrix expressions include terms involving c and the x- and y-direction spacings.The displayed expressions include Δx- and Δy-dependent terms.Source: §C.2 · p. 24
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