Higher Order Runge-Kutta Methods for Fluid Mechanics Problems Abhishek Mishra Graduate Student, Aerospace Engineering Course Presentation MATH 6646

Introduction Problems in fluid mechanics are solved numerically by direct integration of Navier-Stokes equations. These are fully coupled partial differential equations, where additional effects like multiple phases, high turbulence intensities, chemical reactions often complicate the problem. Spatial derivatives can be approximated by standard methods like finite volumes, finite differences etc. The discretization of time-derivative is a difficult choice due to different physical time scales present.

ODE Characterization The discretized ODE is large and stiff, where stiffness can be quantified as ratio of fastest time scale in discretized equations to physical time scale. General solution strategy is to separate ODE into stiff and non-stiff components where f(u) is stiff and g(u) is non-stiff and often nonlinear Implicit-Explicit (IMEX) Runge-Kutta schemes are designed to integrate stiff and non-stiff components separately. For non-separable problems, fully implicit schemes are used.

IMEX Schemes Can be formulated based upon multistage Runge-Kutta (RK) or linear multistep methods (LMM). For separable stiff ODEs, we have Implicit part for stiff component, f(u) Explicit part for non-stiff component, g(u) RK based schemes are made diagonally implicit, so known as DIRK. In general, s-stage diagonally implicit RK scheme can be combined with s+1 (or s) stage explicit RK scheme to give pair of IMEX schemes.

IMEX Schemes Now, we can write general expression for IMEX schemes based on RK methods where u i can be expressed as below,

High Order Time Integrators For non-separable ODE which cannot be decomposed into stiff and non- stiff components, we have to use implicit part of IMEX schemes. High-order implicit schemes can be constructed using multistage or multistep linear schemes. Backward Differentiation Formula (BDF) provides useful approach for multistep schemes. However, BDF based schemes are not L-stable, but just L( α )-stable On the other hand, RK methods are L-stable and can be constructed of any order, p maintaining L-stability.

Implicit Schemes with Order Reduction Higher order implicit schemes generally face order reduction phenomenon. An ODE characterized based upon stiffness parameter, ε can become DAE if ε goes to 0. In such cases, high order scheme depicts lower convergence rates below the classical order of accuracy. Several problems have been studied to test higher order RK schemes over other schemes like BDF.

Problem I: CDR Equations The CDR equations describe conservation of mass, momentum and energy of gas-phase, chemically reacting species. Separable problem: use IMEX partitioning Reaction rates => source of problem stiffness Nonstiff (explicit) Stiff (implicit)

Problem I: CDR Equations Problem exhibits extreme stiffness and non-linearity when large levels of heat release are present. Shock induced combustion waves are simulated.

Convergence Rate Comparisons Comparison of higher order RK type scheme with lower order. Higher order scheme clearly shows better convergence rates for all values of stiffness parameter Test Problem: Van der Pol equation CDR equations

Problem II: Laminar Flow General class of flow problem signifying conservation of mass, momentum energy and turbulent kinetic energy Non-separable ODE: use fully-implicit RK Problem simulates classical case of flow past a circular cylinder at Reynolds number of 1200 and Mach number of 0.3

Timestep and Work Precision The unsteady solution is alternate shedding of vortices from lower and upper surfaces Comparisons of timestep and work (number of iterations) precision show lowest error for 5 th order RK method

Problem III: Turbulent Flow Similar equations as the laminar flow case, significant contribution from turbulent kinetic energy equation, κ Non-separable ODE: use fully-implicit RK Problem simulates flow past a NACA0012 airfoil for Re = 10 3 and Re = 10 5 This test case is chosen to study the effect of Reynolds number on temporal solver.

Problem III: Turbulent Flow Spatial grid is adjusted for each Reynolds number to accommodate changing boundary layer. Both coarse and fine grids are tested as a part of spatial grid refinement study. Flow past NACA0012 airfoil, surface velocities

Timestep Precision Comparison The timestep precision comparison is done with BDFs for both low and high Re number calculations. Clearly at small time step requirements, implicit Runge-Kutta based scheme shows minimum error.

Work Precision Comparison The work precision plot shows anomaly where highest errors occur for RK based 4 th order implicit scheme. This could be due to poor algebraic solver calculations (as reported by [1])

Conclusions The different test problems overall show positive picture for higher order Runge-Kutta schemes. The implicit part of IMEX schemes compute reacting and non-reacting laminar flow problems with high precision. For turbulent flows, at lower accuracy requirements BDF works best. But time precision comparison shows minimum error for RK scheme. Others aspects of the computations like iteration termination strategy, algebraic non-linear solver could also be looked after.

References [1] Carpenter, M. H., Kennedy, C. A., Bijl, H., Viken, S. A., & Vatsa, V. N. (2005). Fourth-order Runge-Kutta schemes for fluid mechanics applications. Journal of Scientific Computing, 25(1-2), [2] Ascher, U. M., Ruuth, S. J., & Spiteri, R. J. (1997). Implicit-explicit Runge- Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25(2), [3] Dimarco, G., & Pareschi, L. (2013). Asymptotic preserving implicit-explicit Runge--Kutta methods for nonlinear kinetic equations. SIAM Journal on Numerical Analysis, 51(2), [4] Jameson, A., Schmidt, W., & Turkel, E. (1981). Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA paper, 1259, 1981.