[Paper Review] On the accuracy of stiff-accurate diagonal implicit Runge-Kutta methods for finite volume based Navier-Stokes equations

Jiawei Wan, Ahsan Kareem+2 more|arXiv (Cornell University)|Jun 5, 2019

Computational Fluid Dynamics and Aerodynamics28 references4 citations

TL;DR

This paper proposes a novel momentum interpolation framework for finite volume discretizations of the incompressible Navier-Stokes equations on collocated grids, enabling stiff-accurate diagonal implicit Runge-Kutta (DIRK) schemes to achieve classical order of accuracy for both velocity and pressure. By treating the semi-discrete system as an index-2 differential-algebraic equation and reformulating momentum interpolation, the method eliminates consistency errors, allowing second- and third-order DIRK schemes to deliver optimal convergence without additional computational cost.

ABSTRACT

The paper aims at developing low-storage implicit Runge-Kutta methods which are easy to implement and achieve higher-order of convergence for both the velocity and pressure in the finite volume formulation of the incompressible Navier-Stokes equations on a static collocated grid. To this end, the effect of the momentum interpolation, a procedure required by the finite volume method for collocated grids, on the differential-algebraic nature of the spatially-discretized Navier-Stokes equations should be examined first. A new framework for the momentum interpolation is established, based on which the semi-discrete Navier-Stokes equations can be strictly viewed as a system of differential-algebraic equations of index 2. The accuracy and convergence of the proposed momentum interpolation framework is examined. We then propose a new method of applying implicit Runge-Kutta schemes to the time-marching of the index 2 system of the incompressible Navier-Stokes equations. Compared to the standard method, the proposed one significantly reduces the numerical difficulties in momentum interpolations and delivers higher-order pressures without requiring additional computational effort. Applying stiff-accurate diagonal implicit Runge-Kutta (DIRK) schemes with the proposed method allows the schemes to attain the classical order of convergence for both the velocity and pressure. We also develop two families of low-storage stiff-accurate DIRK schemes to reduce the storage required by their implementations. Examining the two dimensional Taylor-Green vortex as an example, the spatial and temporal accuracy of the proposed methods in simulating incompressible flow is demonstrated.

Motivation & Objective

• To address the loss of temporal accuracy in Runge-Kutta time integration of incompressible Navier-Stokes equations on collocated grids due to inconsistent momentum interpolation.
• To formally establish the semi-discrete Navier-Stokes system as an index-2 differential-algebraic equation (DAE) under finite volume discretization with momentum interpolation.
• To develop a new momentum interpolation framework that preserves the classical order of convergence for both velocity and pressure when using stiff-accurate DIRK schemes.
• To construct low-storage, stiff-accurate DIRK schemes (SDIRK2 and SDIRK3) that maintain high-order accuracy with minimal memory usage.
• To demonstrate optimal spatial and temporal convergence for velocity and pressure across various boundary conditions and grid types.

Proposed method

• Introduce a new momentum interpolation framework based on a variational formulation that ensures consistency with the index-2 DAE structure of the semi-discrete Navier-Stokes system.
• Reformulate the time integration of the DAE system using stiff-accurate diagonal implicit Runge-Kutta (DIRK) schemes, ensuring that both velocity and pressure achieve classical order of accuracy.
• Develop two families of low-storage DIRK schemes: one requiring 2 velocity and 1 pressure register, the other 3 velocity and 1 pressure register, to reduce memory consumption.
• Implement a solution algorithm using Picard linearization for nonlinearity and PISO for pressure-velocity coupling, ensuring discrete continuity enforcement at each internal stage.
• Use the Taylor-Green vortex on structured Cartesian grids as a benchmark to verify spatial and temporal accuracy under various boundary conditions.
• Validate the method against the standard approach, showing that the proposed framework achieves second-order spatial convergence and classical temporal convergence, while the standard method fails for unsteady boundary conditions.

Experimental results

Research questions

• RQ1 Does the standard momentum interpolation in finite volume methods on collocated grids preserve the classical order of accuracy for both velocity and pressure when using stiff-accurate DIRK schemes?
• RQ2 How does the differential-algebraic structure (index-2) of the semi-discrete Navier-Stokes system affect the convergence behavior of Runge-Kutta time integrators?
• RQ3 Can a new momentum interpolation framework be designed to restore classical order of accuracy for pressure in time integration, even under time-varying boundary conditions?
• RQ4 What is the impact of the momentum interpolation scheme on the temporal convergence of the pressure field in implicit Runge-Kutta time integration?
• RQ5 Can low-storage DIRK schemes be constructed such that they maintain high-order accuracy and stability while minimizing memory usage in finite volume simulations?

Key findings

• The proposed momentum interpolation framework ensures that both velocity and pressure achieve second-order spatial convergence for both GFISDM-Z and GFISDM-H schemes on structured grids.
• When using the proposed method with SDIRK2 and SDIRK3 schemes, the temporal solutions for velocity and pressure achieve the classical order of convergence (second and third order, respectively), even under unsteady Dirichlet boundary conditions.
• In contrast, the standard approach fails to achieve second-order temporal accuracy for pressure under unsteady boundary conditions, resulting in only first-order convergence.
• SDIRK3 outperforms SDIRK2 in terms of accuracy per computational cost when sufficient Picard iterations are used, especially for problems with strong nonlinearity.
• The proposed method maintains optimal convergence even when Picard iterations are terminated early, due to the enforcement of the discrete continuity equation at each internal stage.
• The analysis is valid for static, structured Cartesian and curvilinear grids using polynomial-based interpolation; the results do not extend to deforming meshes without additional enforcement of geometric conservation laws.