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Numerical Approximation of Incompressible Navier-Stokes Equations Based on an Auxiliary Energy Variable
Lianlei Lin, Suchuan Dong
TL;DR
The paper addresses the computational cost of energy-stable incompressible Navier–Stokes discretizations while retaining stability advantages. It introduces a scalar auxiliary energy variable and an efficient decoupled algorithm, which exhibits second-order temporal and exponential spatial convergence and supports stable large-time-step simulations for steady flows.
Problem
Energy-stable schemes can reduce time-step restrictions but typically require expensive nonlinear or variable-coefficient solves, whereas low-cost semi-implicit schemes are only conditionally stable.
Method
The method reformulates the incompressible Navier–Stokes equations using a scalar auxiliary variable governed by a dynamic equation, then solves decoupled constant-coefficient linear systems plus a scalar nonlinear equation.
Results
The method shows second-order convergence in time and exponential convergence in space for smooth solutions, while producing stable results with large time steps for steady-flow computations.
Takeaways & Limitations
The auxiliary-variable formulation combines a discrete modified-energy stability property with an efficient implementation based on decoupled solves and a negligible-cost scalar nonlinear iteration.
Takeaways & Limitations
The formulation assumes prescribed boundary velocity and a compatible initial velocity satisfying the governing constraints.
Abstract
from arXiv · showhide
We present a numerical scheme for approximating the incompressible Navier-Stokes equations based on an auxiliary variable associated with the total system energy. By introducing a dynamic equation for the auxiliary variable and reformulating the Navier-Stokes equations into an equivalent system, the scheme satisfies a discrete energy stability property in terms of a modified energy and it allows for an efficient solution algorithm and implementation. Within each time step, the algorithm involves the computations of two pressure fields and two velocity fields by solving several de-coupled individual linear algebraic systems with constant coefficient matrices, together with the solution of a nonlinear algebraic equation about a {\em scalar number} involving a negligible cost. A number of numerical experiments are presented to demonstrate the accuracy and the performance of the presented algorithm.
1 Introduction
The paper develops an incompressible Navier–Stokes scheme aimed at balancing unconditional energy stability with computational efficiency. It introduces a scalar auxiliary energy variable and an implementation based on decoupled constant-coefficient solves.
- 1 Introduction: The approach is motivated by applications requiring efficient incompressible-flow algorithms, including simulations related to two-phase and multiphase flows.The paper notes that analogous mathematical structures make such algorithms relevant beyond single-phase incompressible flows.
- 1 Introduction: The method targets the tradeoff between energy-stable schemes with high computational cost and low-cost semi-implicit schemes restricted by conditional stability.Energy-stable methods can permit larger time steps, whereas semi-implicit methods are popular because of their lower cost but may face CFL-type restrictions.
- 1 Introduction: The scheme introduces a scalar auxiliary variable associated with total system energy and reformulates the Navier–Stokes equations into an equivalent system.This reformulation is the basis for the numerical method and its implementation.
- 1 Introduction: Each time step computes two pressure fields and two velocity fields through linear systems with pre-computable constant coefficient matrices.The remaining nonlinear solve concerns only a scalar number and costs approximately 1–2% of each time step.
- 1 Introduction: The paper presents the reformulation, numerical scheme, implementation strategy, and numerical tests of accuracy and performance.The numerical simulations are presented in Section 3, following the formulation and implementation discussion.
2 Auxiliary Variable-Based Algorithm for Incompressible Navier-Stokes Equations
The paper reformulates incompressible Navier–Stokes equations using a scalar auxiliary variable tied to shifted total energy, yielding an energy-stable scheme and an efficient decoupled implementation.
- Reformulated Equations: The auxiliary variable is a scalar associated with shifted total energy, and the reformulated system is equivalent to the original Navier–Stokes equations.A constant C0 is chosen so the shifted energy remains positive, and both energy-related variables are scalar quantities rather than fields.
- Numerical Scheme: The scheme uses a J-th order backward differentiation formula with explicit convection treatment and computes each new state from the previous velocity, pressure, and auxiliary variable.The temporal order is J = 1 or 2, with Δt denoting the time-step size.
- Energy Stability: In the absence of external forcing and with zero boundary velocity, the scheme satisfies a discrete energy stability property.The stability result follows by combining the velocity and auxiliary-variable relations under the stated boundary and forcing conditions.
- Implementation: An explicit boundary-vorticity approximation slightly reduces stability robustness while simplifying implementation and reducing computations.The approximation replaces the unknown boundary vorticity contribution with one based on an explicitly approximated velocity.
- Solution Algorithm: The pressure and velocity equations are decoupled through separate auxiliary pressure and velocity problems, while the scalar S determines their final combinations.The implementation uses constant-coefficient linear systems for the field variables and postpones determination of S to a scalar equation.
- Implementation: The time-step algorithm solves two pressure fields and two velocity fields, then obtains S from a nonlinear scalar equation using Newton’s method at negligible cost.Once S is known, the new energy, auxiliary variable, velocity, and pressure are recovered from the corresponding relations.
3 Representative Numerical Examples
Numerical tests show accurate convergence, strong stability at large time steps for steady Kovasznay flow, and physically consistent cylinder-flow results within tested settings.
- 3.1 Convergence Rates: Second-order temporal convergence is observed as ∆t decreases from 0.1 to 1.953125×10−4 with element order 16.The comparison uses L∞ and L2 errors for velocity and pressure.
- 3.2 Kovasznay Flow: Steady-state Kovasznay velocity errors decrease exponentially with element order before saturating near 10−13 at order 14 and above.These tests use ∆t = 0.005 and C0 = 0.01.
- 3.2 Kovasznay Flow: For Kovasznay flow, increasing C0 to 500 leaves the error near 10−4, while C0 = 0.01 reaches errors between 10−14 and 10−13.The tests use ∆t = 0.005 and element order 14.
- 3.2 Kovasznay Flow: Compared with the semi-implicit scheme, the current method remains stable at much larger ∆t values, whereas the baseline blows up beyond about 0.01.The comparison tracks the L2 norm of the x velocity.
- 3.2 Kovasznay Flow: The current scheme remains stable for Kovasznay flow with time steps as large as ∆t = 1000, although accuracy is not expected at such sizes.The tests use element order 10 and C0 = 0.01.
- 3.3 Flow Past a Circular Cylinder in a Periodic Channel: In periodic-cylinder simulations, computed mean drag agrees with the imposed driving force to within 1% across tested driving forces and Reynolds numbers.The mean lift is zero or essentially zero, with small rms lift.
4 Concluding Remarks
The auxiliary-variable scheme achieves discrete energy stability while retaining an efficient implementation based on decoupled constant-coefficient linear systems and a negligible scalar nonlinear solve.
- 4 Concluding Remarks: The scheme exhibits discrete energy stability in terms of a modified energy and uses decoupled pressure and velocity computations.Each time step computes two pressure variables and two velocity variables before updating the scalar auxiliary variable.
- 4 Concluding Remarks: The pressure and velocity solves use constant-coefficient Poisson and Helmholtz systems, while Newton’s scalar solve costs roughly 2% per time step.The nonlinear equation is posed for a scalar number rather than a field function.
- 4 Concluding Remarks: The method achieves second-order temporal convergence and exponential spatial convergence for smooth solutions.The algorithm is implemented with a C0 spectral element technique.
- 4 Concluding Remarks: Large time steps produce stable computations for steady-flow problems, although accurate flow fields require time steps that are not too large.The conclusion distinguishes stability at large time steps from accuracy of the computed flow field.
- 4 Concluding Remarks: Compared with other energy-stable Navier-Stokes schemes, the algorithm uses time-independent coefficient matrices and offers an alternative to semi-implicit production schemes.This combines an energy stability property with an efficient implementation strategy.