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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

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

Ultimately, the Navier-Stokes equations can be

written as

Compressible Incompressible

Autonomous

System of ODEs System of DAEs

Non-autonomous

Rationale for Higher Order Integration Method

(Explicit)

Global error for a p-th order time integration

Define work function

where

(No Transcript)

Real Example Dormand and Prince (1986) - When

accuracy is of highest priority, high-order

method is more efficient

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

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.

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.

One-Step Methods for a System of ODEs

Runge-Kutta Method - 1

Carl David Tolmé Runge (1856-1927)

Martin Wilhelm Kutta (1867-1944)

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)

Runge-Kutta Method - 3

Estimating the integral

Forward Euler (1st order)

Backward Euler (1st order)

Trapezoidal (2nd order)

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

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

Runge-Kutta Method - 6

Generalized Runge-Kutta Formulation

An s-stage R-K for the ODE system

Butcher array

(row sum condition)

If explicit

Runge-Kutta Method - 7

Examples of Butcher Array

Forward Euler

2nd order R-K (one parameter family)

Classical 4th order R-K

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.

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

Runge-Kutta Method - 10

Stability Boundaries of Explicit R-K Methods

p4

p3

p2

p1

(Ref. 1)

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.

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)

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)

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

Multistep Methods for a System of ODEs

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

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)

Multistep Method 3

Adams-Bashforth Family Explicit

Forward Euler

AB2

Multistep Method 4

Adams-Moulton Family Implicit

Backward Euler

Trapezoidal

Multistep Method 5

BDF (Backward Differentiation Formulas) Gear

Method

- Implicit, stable and accurate - Usually

implemented with modified Newton method for

nonlinear systems.

Backward Euler

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)

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)

Numerical Methods for a System of DAEs

System of DAEs

Example Incompressible Navier-Stokes Equations

Define differential variables

algebraic variables

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

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

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.

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).

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