Direct Numerical Simulations of Multiphase Flows

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Higher Order Time Integration for Ordinary
Differential Equations and Differential
Algebraic Equations
Instructor Hong G. Im University of Michigan
Fall 2005
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Outline

• Rationale for higher order methods
• Stiffness - definition
• One-step (multi-stage) methods for ODEs
• - The classical 4th order Runge-Kutta method
• - General R-K methods
• - Error control
• - Stiffness and implicit methods
• Multi-step methods for ODEs
• - Adams methods
• - Backward differentiation formulas (BDF)
• Higher order methods for DAEs
• - Hessenberg form
• - Index

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Ultimately, the Navier-Stokes equations can be
written as
Compressible Incompressible
Autonomous
System of ODEs System of DAEs
Non-autonomous
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Rationale for Higher Order Integration Method
(Explicit)
Global error for a p-th order time integration
Define work function
where
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(No Transcript)
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Real Example Dormand and Prince (1986) - When
accuracy is of highest priority, high-order
method is more efficient
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Stiffness - 1
- Observation Consider a system of ODEs
If integrating by explicit Euler method, the
system is stable if for all which is
an eigenvalue of matrix
- Large eigenvalue means Solution varies
rapidly in time. Time step has to be
small a) to resolve the temporal variation b)
to keep the explicit method stable
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Stiffness - 2
Definition of Stiffness
- Mathematical Definition A system of ODE is
called stiff when the magnitudes of the
eigenvalues cover a wide range. - Physical
Definition An initial value problem is called
stiff when the physical processes have widely
varying time scales (e.g. chemically reacting
systems) - Practical Definition An initial
value problem is stiff if the time step size
needed to maintain stability of the forward Euler
method is much smaller than the step size
required to represent the solution
accurately.
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Stiffness - 3
Definition of A-Stability
- An ODE solution method is A-stable if its
region of absolute stability contains the
entire left half-plane of z??t (Re(z)
? 1) - A-stability is an important property
to have for a numerical method dealing with
stiff problems.
Definition of L-Stability
- If a method is A-stable and the stability
function (such as yn1/yn) vanishes as z ?
???, the method is L-stable.
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One-Step Methods for a System of ODEs
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Runge-Kutta Method - 1
Carl David Tolmé Runge (1856-1927)
Martin Wilhelm Kutta (1867-1944)
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Runge-Kutta Method - 2
The Runge-Kutta Method
Higher-order accuracy is achieved by several
function evaluations (stages)
Consider an initial value problem (can be easily
extended to a system of ODEs)
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Runge-Kutta Method - 3
Estimating the integral
Forward Euler (1st order)
Backward Euler (1st order)
Trapezoidal (2nd order)
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Runge-Kutta Method - 4
To make the trapezoidal method explicit,
approximate
by the forward Euler method
Predictor
Corrector
2nd order Runge-Kutta method explicit
trapezoidal method in two stages
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Runge-Kutta Method - 5
Classical 4th order Runge-Kutta method
Using the Simpsons quadrature formula
and evaluate by forward Euler
4-stage, 4th-order R-K
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Runge-Kutta Method - 6
Generalized Runge-Kutta Formulation
An s-stage R-K for the ODE system
Butcher array
(row sum condition)
If explicit
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Runge-Kutta Method - 7
Examples of Butcher Array
Forward Euler
2nd order R-K (one parameter family)
Classical 4th order R-K
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Runge-Kutta Method - 8
Designing R-K Method
Order conditions for p-th order R-K
where
for each , order conditions for
- In general, the order conditions are not
sufficient - Usually - The remaining degrees of
freedom can be used to improve stability, etc.
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Runge-Kutta Method - 9
Stability of R-K Method
Linear stability condition based on
For
Applying to the classical 4th order R-K
Stability Condition
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Runge-Kutta Method - 10
Stability Boundaries of Explicit R-K Methods
p4
p3
p2
p1
(Ref. 1)
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Runge-Kutta Method - 11
Linear Stability of R-K Method General
For
where the Butcher coefficients
Kronecker delta
- For explicit method - Implicit methods allow
for A-stability - If
then the implicit R-K (IRK) is L-stable.
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Runge-Kutta Method - 12
Further Remarks on Stability of R-K Methods
- For an s-stage method with order p lt s, the
absolute stability depend on the methods
coefficients. - No explicit R-K method has an
unbounded region of absolute stability.
Since as ,
very large negative values of cannot be
in the region of absolute stability.
(A-stability)

• Explicit R-K (ERK) is inappropriate for stiff
  problems
• Implicit R-K (IRK), Additive R-K (ARK)

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Runge-Kutta Method - 13
Error Estimation and Time Step Control
Embedded R-K Method - Runs a pair of R-K of
orders p and p1.
User-specified tolerance
(I)
(PI)
(PID)
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Runge-Kutta Method 14
Additive Runge-Kutta (ARK) Method
For a system of separably stiff ODEs
Nonstiff terms (convection, diffusion)
Stiff terms (reaction, diffusion)
ARK ERK (for nonstiff) IRK (for stiff)
- IRK requires Jacobian evaluation, iteration. -
Coupling conditions to maintain accuracy (ERK and
IRK) - Diagonally-implicit (DIRK) when stiff
terms are local
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Multistep Methods for a System of ODEs
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Multistep Method 1
One-step method high accuracy achieved by many
function evaluations Multistep
method high accuracy achieved by many
prior time step solutions
A simplest example Leapfrog method
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Multistep Method 2
General form of a k-step linear multistep method
- Without loss of generality,
- Method is explicit if
- Method is called linear because RHS is linear
in f Does not mean that f is a linear
function of y and t.
- As a consequence, the local truncation error
always has the simple expression (for p-th order)
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Multistep Method 3
Adams-Bashforth Family Explicit
Forward Euler
AB2
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Multistep Method 4
Adams-Moulton Family Implicit
Backward Euler
Trapezoidal
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Multistep Method 5
BDF (Backward Differentiation Formulas) Gear
Method
- Implicit, stable and accurate - Usually
implemented with modified Newton method for
nonlinear systems.
Backward Euler
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Multistep Method 6
Stability of Multistep Methods
k3
k4
k3
k1
k2
k2
k4
Adams-Bashforth k1,2,3,4 Adams-Moulton
k2,3,4
(Ref. 1)
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Multistep Method 6
Stability of Multistep Methods (Stable OUTSIDE
shaded area)
k3
k6
k5
k2
k4
k1
BDF k1,2,3 BDF
k4,5,6
(Ref. 1)
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Numerical Methods for a System of DAEs
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System of DAEs
Example Incompressible Navier-Stokes Equations
Define differential variables
algebraic variables
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Index of DAEs
For a DAE system
The index is 1 if is nonsingular.
Special DAE Forms
Hessenberg Index-1 Hessenberg Index-2
Incompressible N-S Equation
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Softwares

• Runge-Kutta Method
• Nonstiff Problems RFK45, DOPRI5, ODE45
  (Matlab 5)
• Stiff Problems RADAU5, STRIDE
• Multistep Method
• Nonstiff Problems LSODE, VODE
• Stiff Problems DIFSUB, VODPK
• DAE Systems DASSL, DASPK, LIMEX
• Compiled from Ref. 1

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References - 1
Monographs

• Ascher, U.M. and Petzold, L.R., Computer Methods
  for Ordinary Differential Equations and
  Differential-Algebraic Equations, SIAM, 1998.
• Brenan, K.E., Campbell, S.L, and Petzold, L.R.,
  Numerical Solution of Initial-Value Problems in
  Differential-Algebraic Equations, SIAM, 1996.
• Hairer, E., Norsett, S.P., and Wanner, G.,
  Solving Ordinary Differential Equations I
  Nonstiff Problems., Springer-Verlag, 2nd Ed.,
  1993.
• Hairer, E. and Wanner, G., Solving Ordinary
  Differential Equations II Stiff and
  Differential-Algebraic Problems, 2nd ed.,
  Springer-Verlag, 1996.
• Dormand, J.R., Numerical Methods for Differential
  Equations A Computational Approach, CRC Press,
  1996.

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References - 2
Journal Papers

• Prince, P.J. and Dormand, J.R., High order
  embedded Runge-Kutta formulae, Journal of
  Computational and Applied Mathematics, 7 67
  (1981).
• Dormand, J.R. and Prince, P.J., A reconsideration
  of some embedded Runge-Kutta formulae, Journal of
  Computational and Applied Mathematics, 15
  203-211 (1986).
• Sharp, P.W., Numerical comparisons of some
  explicit Runge-Kutta pairs of orders 4 through 8,
  ACM Transactions on Mathematical Software, 17
  387-409 (1991).
• Kennedy, C.A. and Carpenter, M.H., Several new
  numerical methods for compressible shear-layer
  simulations, Applied Numerical Mathematics, 14
  397-433 (1994).
• Kennedy, C.A., Carpenter, M.H., and Lewis, R.M.,
  Low-storage,explicit Runge-Kutta schemes for the
  compressible Navier-Stokes equations, Applied
  Numerical Mathematics, 35 177-219 (2000).
• Kennedy, C.A. and Carpenter, M.H., Additive
  Runge-Kutta schemes for convection-diffusion-react
  ion equations, Applied Numerical Mathematics, 44
  139-181 (2003).