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An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations
Floraine Cordier, Pierre Degond, Anela Kumbaro
TL;DR
The paper addresses the difficulty of simulating flows spanning compressible and low-Mach incompressible regimes. It develops a semi-implicit AP scheme that treats acoustic effects implicitly, yielding consistent all-speed behavior with Mach-independent time steps, while numerical tests validate the approach within stated scope limits.
Problem
Compressible-flow solvers face increasingly restrictive time steps as the Mach number decreases, yet simulations must handle both compressible and incompressible regimes.
Method
The scheme treats acoustic terms implicitly and convective and diffusive terms explicitly, using an elliptic pressure equation within a conservative AP discretization.
Results
The second-order discretization shows good behavior in both regimes, with consistent incompressible limiting behavior, divergence-free velocity, good shock properties, and Mach-independent time steps.
Takeaways & Limitations
The scheme provides an all-speed method for mixed compressible and incompressible flows, including steady or unsteady cases and general equations of state.
Takeaways & Limitations
The intended second-order-in-time scheme could not be used because fully implicit pressure treatment generated spurious oscillations; “second-order” therefore means first-order time and second-order space.
Abstract
from arXiv · showhide
We present an Asymptotic-Preserving 'all-speed' scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretization, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime. The divergence-free condition on the velocity in the incompressible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and Two-dimensional numerical results provide a validation of the Asymptotic-Preserving 'all-speed' properties.
1 Introduction
The paper develops an Asymptotic-Preserving all-speed scheme for mixed compressible and incompressible flows, addressing low-Mach numerical difficulties while retaining compressible-flow behavior. Its design targets ε-independent stability and consistency with the incompressible limit.
- Motivation: All-speed methods must handle both order-unity-Mach compressible regions and very-low-Mach incompressible regions in the same flow.As the Mach number tends to zero, compressible Euler and Navier–Stokes equations converge to their incompressible counterparts.
- Asymptotic-Preserving framework: The AP property requires ε-independent stability and a discretized ε→0 limit that consistently approximates the continuous limit model.The continuous model Mε represents compressible flow, while M0 represents incompressible Euler or Navier–Stokes flow.
- Present approach: The proposed scheme targets mixed flows and explicitly satisfies the incompressible divergence-free condition while preserving compressible-regime shock speeds.It is also designed for general equations of state and both steady and unsteady flows.
- Present approach: The method extends earlier AP work from isentropic Euler equations to full Euler and Navier–Stokes equations with energy conservation and general equations of state.The extension requires changing which terms are treated implicitly and handling a nonlinear pressure equation for real-gas equations of state.
- Motivation: Low-Mach regions are difficult to simulate because explicit compressible schemes face time steps proportional to the Mach number.The acoustic speed scales as 1/ε, making the CFL restriction increasingly severe as ε decreases.
- Present approach: Unlike prior pressure-based approaches, the scheme uses a single conservative step and solves only a simple elliptic equation implicitly.This reduces computational cost relative to predictor-corrector procedures while retaining the pressure-equation strategy.
2 Time semi-discrete scheme
The semi-discrete scheme treats the full Euler and Navier–Stokes equations with a semi-implicit formulation that enables an elliptic pressure solve and supports general equations of state.
- Scope and formulation: The Navier–Stokes formulation includes density, momentum, energy, viscosity, heat conduction, and external-force terms.The scheme is presented on the full Euler equations first, with Navier–Stokes right-hand terms added explicitly.
- Semi-discrete design: The time discretization treats internal energy implicitly while evaluating kinetic energy explicitly, with α≈1 used to prevent strong-shock oscillations.The parameter α is independent of the Mach number.
- Semi-discrete design: The density is advanced through an explicit continuity equation, while momentum and pressure-related energy terms are treated implicitly.This arrangement is central to constructing the elliptic pressure equation.
- Pressure solve: Substituting the momentum equation into the energy equation produces an elliptic pressure equation for perfect-gas and general equations of state.For a general EOS, the pressure and enthalpy are solved together subject to the updated density.
- Pressure solve: After solving pressure and energy variables, the updated momentum is obtained from the momentum equation, whose terms remain O(1) in the low-Mach regime.The elliptic equation controls the pressure scaling needed for this property.
- Scope and formulation: The scheme extends earlier isentropic AP schemes to the full Euler and Navier–Stokes equations, including energy.This extension changes how the pressure equation is constructed.
3 Full time and space discretization
The fully discrete finite-volume scheme combines explicit convective treatment with implicit pressure coupling, yielding a Mach-number-independent stability time step and a second-order spatial variant.
- First-order discretization: The first-order finite-volume discretization applies on structured or unstructured multidimensional meshes and uses centered discretization for implicit pressure.The scheme includes general source terms such as gravity and Navier–Stokes diffusion.
- Stability: The CFL time step depends only on fluid velocity and not on the Mach number because acoustic terms are treated implicitly.This avoids the explicit-method restriction caused by the acoustic velocity diverging as Mach number decreases.
- Flux treatment: Lax–Friedrichs upwinding is applied only to the explicit flux components, while the implicit pressure gradient does not affect stability.The explicit-part eigenvalues use the fluid-normal velocity and sound speed.
- Resolution: The discrete system is solved by advancing density explicitly, solving an elliptic pressure problem, and then updating momentum and velocity.Perfect-gas pressure solves are linear, whereas general-EOS solves use Newton iteration for pressure and enthalpy.
- Second-order scheme: The second-order spatial scheme reconstructs limited states with a minmod limiter to reduce diffusion while avoiding spurious oscillations.The time discretization remains first order in the reported second-order scheme.
- Second-order scheme: A fully second-order RK2CN time-space scheme could not be used because fully implicit pressure generated spurious oscillations that remained unresolved.The reported “second-order scheme” is therefore first order in time and second order in space.
4 Asymptotic preserving property
The asymptotic analysis shows that the scheme converges consistently to the incompressible limit as Mach number vanishes and enforces the discrete divergence constraint to O(∆t).
- Continuous limit: The continuous low-Mach analysis derives the incompressible limit for full Euler equations with a general equation of state.Unlike the isentropic case, the velocity divergence constraint comes from the energy equation.
- Continuous limit: In the limit, pressure separates into spatially uniform thermodynamic pressure and dynamic pressure linked to fluid motion.The thermodynamic pressure is assumed independent of time under the chosen boundary conditions.
- Continuous limit: The low-Mach mass equation implies that density is constant along each fluid-element trajectory, rather than necessarily constant in space.This differs from the isentropic low-Mach limit described in the analysis.
- Discrete limit: The semi-discrete scheme is consistent with the limiting system, establishing its asymptotic-preserving property.The fully discrete proof follows the same methodology.
- Discrete limit: The discrete limit system is constructed by expanding the semi-discrete scheme in powers of ε and retaining leading-order and second-order momentum equations.The resulting system is compared with the continuous incompressible model.
- Discrete incompressibility: The discrete velocity satisfies the incompressibility constraint ∇·u^n+1=O(∆t), preventing accumulation of an O(1) divergence over time steps.The proof directly establishes the one-step estimate rather than relying only on the limit velocity.
5 Numerical results
Numerical tests assess the second-order scheme on acoustic pulses, compressible shock tubes, and low-Mach incompressible flows. The results show second-order spatial convergence for smooth pulses, satisfactory shock behavior, and agreement with a low-Mach reference where classical Roe fails.
- Colliding acoustic pulses: The colliding-pulses experiment uses a one-dimensional periodic domain with Mach number ε = 1/11 and 220 cells.The time-step is ∆t = 1 × 10−3s and the gas has γ = 1.4.
- Colliding acoustic pulses: The colliding-pulses test tracks pressure during pulse superposition at t = 0.815s and separation at t = 1.63s.Weakly nonlinear effects distort the final profile as shocks begin forming near x = ±18.5.
- Colliding acoustic pulses: The pressure solution demonstrates second-order convergence in space when the time-step scales as ∆t = 0.05 × ∆x2.Errors are measured against a 3200-cell reference using 100, 200, and 400 cells.
- Shock tube problems: In the Sod shock tube, the second-order scheme has small overall deviation from the exact solution and satisfactory shock velocities.Discontinuities reduce the spatial convergence order from 2 to 1.
- Shock tube problems: In the stronger-shock Lax problem, accuracy is satisfactory and shock velocities are accurately restored, with discontinuities again reducing spatial convergence from 2 to 1.The errors are compared with the exact solution for 100, 200, and 400 cells.
- Low-Mach tests: For the low-Mach recirculation test, the AP scheme computes a circulation region similar to the Low Mach Roe scheme, whereas classical Roe captures no recirculation.The computed solution also matches the low-Mach reference for local Mach numbers ranging from 10−5 to 10−9.
6 Conclusion
The proposed scheme targets mixed compressible–incompressible flows and performs well across both regimes, including weakly compressible test cases. Initial mixture-model tests also support its use with general equations of state, heat transfer, and phase change, while more elaborate two-phase models remain future work.
- The second-order scheme performs well in compressible and low Mach-number regimes, preserving shock properties, incompressible consistency, divergence-free velocity, and elliptic pressure resolution.
- Mach-number-independent time steps can be based on fluid velocity rather than acoustic velocity, allowing larger steps than explicit upwind methods.
- Weakly compressible backward-facing-step and lid-driven-cavity tests reproduce expected fluid recirculations, while the heat-driven cavity gives the correct solution using the energy equation.
- More elaborate four-equation mixture and six-equation two-fluid models are identified as future extensions.
- Initial four-equation mixture-model tests include heated-channel water flow and confirm capability with a general water-and-steam equation of state, heat transfer, and phase change.