[Paper Review] Error Estimate of MacCormack Rapid Solver Method for 2D Incompressible Navier-Stokes Problems

Eric Ngondiep|arXiv (Cornell University)|Mar 20, 2019

Advanced Numerical Methods in Computational Mathematics4 citations

TL;DR

This paper analyzes the error estimates and convergence rate of a two-level MacCormack Rapid Solver (MCRS) method for 2D incompressible Navier-Stokes equations, combining explicit MacCormack and Crank-Nicolson schemes. Theoretical and numerical results confirm second-order accuracy in both time and space, with convergence rates of approximately 2.0 for velocity and pressure errors under uniform refinement.

ABSTRACT

The error estimates and convergence rate of a two-level MacCormack rapid solver method for solving a two-dimensional incompressible Navier-Stokes equations are analyzed. This represents a continuation of the work on the stability analysis of the method. The theoretical result suggests that the rapid solver method is both convergent and second order accurate with respect to time step $\\Delta t.$ A wide set of numerical evidences confirm this theoretical analysis.

Motivation & Objective

• To analyze the error estimates and convergence rate of a two-level MacCormack Rapid Solver (MCRS) method for 2D incompressible Navier-Stokes equations.
• To extend prior stability analysis by providing global error estimates and convergence rate validation.
• To demonstrate the efficiency and robustness of the MCRS scheme compared to first-order methods like the two-level finite element Galerkin approach.
• To confirm theoretical findings through extensive numerical experiments with varying mesh size and time step.

Proposed method

• The MCRS method combines an explicit predictor-corrector MacCormack scheme with a Crank-Nicolson time discretization for solving the 2D incompressible Navier-Stokes equations.
• A two-level formulation is employed, where the MacCormack method handles the nonlinear convective terms explicitly, and the Crank-Nicolson scheme ensures second-order accuracy in time for the viscous and pressure terms.
• The method uses a variational formulation in space, with finite element discretization over a triangulated domain, and enforces incompressibility via a mixed finite element approach.
• Error norms are defined in the L²(0,T;L²) and L²(0,T;H¹) spaces for velocity, pressure, and its gradient to quantify convergence.
• Convergence rates are computed using the formula $ r^{1}_{( ext{cdot})} = \log_2\left( \frac{E^{N}((\text{cdot}))}{E^{N-1}((\text{cdot}))} \right) $, with $ \Delta t = h $, to assess order of accuracy.
• Numerical tests are conducted on two benchmark problems with exact solutions, using uniform mesh refinement and time step reduction.

Experimental results

Research questions

• RQ1 Is the two-level MCRS method second-order accurate in time and space for the 2D incompressible Navier-Stokes equations?
• RQ2 What is the global convergence rate of the MCRS method in terms of velocity, pressure, and velocity gradient errors?
• RQ3 How does the MCRS method compare in accuracy and efficiency to first-order methods like the two-level finite element Galerkin scheme?
• RQ4 To what extent do numerical experiments confirm the theoretical error estimates and convergence rates?
• RQ5 Does the hybrid MacCormack–Crank-Nicolson approach maintain stability and accuracy over long time intervals?

Key findings

• Theoretical analysis confirms that the MCRS method is second-order accurate in time step $ \Delta t $, with convergence rate approaching 2.0 for velocity and pressure errors.
• Numerical experiments show that the convergence rate for velocity $ u $ is approximately 2.0000 at $ \Delta t = 2^{-7} $, with $ r^{1}_u = 2.0000 $, indicating second-order convergence.
• For pressure $ p $, the convergence rate stabilizes at around 2.0000 for $ \Delta t \leq 2^{-7} $, confirming second-order accuracy.
• The gradient error $ E^N(\nabla u) $ converges at a rate of approximately 1.415, consistent with the expected order for H¹-norm errors.
• The method outperforms the first-order two-level finite element Galerkin approach in both accuracy and computational efficiency.
• The MCRS scheme demonstrates robustness and reliability across different Reynolds numbers and problem configurations, as validated by tests with exact solutions.