Solving Constrained Differential- Algebraic Systems Using Projections. Richard J. Hanson Fred T. Krogh August 16, mathalacarte.

Transcription

1 Solving Constrained Differential- Algebraic Systems Using Projections Richard J. Hanson Fred T. Krogh August 6, mathalacarte.com

2 Abbreviations and Terms ODE = Ordinary Differential Equations DAE = Differential-Algebraic Equations PDE = Partial Differential Equations BDF = Backward Differentiation Formulas Index of a DAE

3 Abbreviations and Terms DASSL, DASSLX, DASPG, DASPK Names of Fortran 77 software for solving DAE and ODE systems. They all originated with work of Linda Petzold and colleagues.

4 Agenda DAE Problem Statements Noting Constraint Violations Outline of BDF Solution for DAE DAE Problem Statement with Constraints Projection following BDF Step Examples using Projection Method Issues about the software

5 DAE Problem Statement yn F N ( ) = ( ) y t ( ) , a dynamic vector, a vector function: dy F( t, y, y ) = 0, y =, dt y t = y y t = y

6 A Sampling of Applications Modeling dynamic systems Financial Engineering Solving PDE systems

7 Planar Pendulum In rectangular coordinates: ( x y L ) 2 d x x = λx, x =, 2 dt y = λy g 2 + = 0

8 Pendulum as First Order System Define components of a vector and write the model in these symbols: y y y y y = = = = = x y y y λ y y = 2 3 y y = 2 4 yy y = 0 yy+ y + g= ( y ) y2 L 0 + = 0

9 Make a Note: Kinetic energy + Potential energy = Constant, Plus initial conditions: ( 2 2 y ) 3 + y4 + y2g = 0 2 y 0 = L, y 0 = 0, j = 2,3,4,5 ( ) ( ) j

10 Reduction of Index, a source of constraint violation This constraint (the length) is differentiated three or two times to yield an index 0 or index problem: ( 2 2 2) 2 2 y + y L = 0

11 Ancillary Constraints from Index Reduction ( 2 2 2) 2 2 y + y L = yy yy yy yy 0 (An original system equation) = = 0 d ( yy yy ) yl y y gy dt = = 0 d dt ( ) yl + y + y gy = yl 3yg=

12 Dynamic Length of Pendulum?

13 Constrained DAE Problem yn F N ( ) = ( ) y t ( ) , a dynamic vector, a vector function: dy F( t, y, y ) = 0, y =, dt y t = y y t = y Gty (, ) = 0, Mconstraint functions

14 Constraints for Pendulum ( 2 2 2) 2 0, Constraint is a system equation 2 y + y L = yy + yy = 0, Constraint with index and index 0 systems yl + y + y gy = , Constraint with index 0 system ( 2 2) y + y + y g = 0, Total Energy, index and index 0 systems

15 Modifications to DASSL Recall the predictor-corrector steps used to solve index or index 0 systems using DASSL:. Predict yn, y + n+ using interpolation of known values yn i, i = 0,, k 2. Correct by solving ( α β) F t, y, y+ = 0 for y = y n+ n+ 3. We have a candidate that may violate the constraints G( tn+, yn+ ) = 0

16 Moving to the Constraints. We intervene before accepting 2. Solve Gt ( y dy) 0 dy = for n+, n+ 3. Use Newton s method: y n + 4. Accept the projected step, with a possible scaling for size: y = y dy n+ n+ ( ) Cdy G t y C G y = n+, n+, =

17 Some Details of the Newton Step. Compute the minimum length solution of (, ) Cdy = G t y n+ n+ 2. Use a norm define by the accuracy or tuning weights WT = RTOL y + ATOL, i =,, N i i i i 3. Move onto the constraints with a step that N /2 2 /2 minimizes WT du, dy = W du i= i i

18 Comments on the Linear Algebra. We use plane rotations to solve the system: ( ) CW du = G t, y, dy = W du /2 /2 n+ n+ 2. At each step the constraint system is computed, factored and solved. 3. For large sparse problems one could solve the square system A 0 u b = = = T I A v 0 ( ) /2, A CW, b G tn+, yn+

19 Examples, Pendulum Problem The following slides will show that:. Using constraints and the projection method one can achieve acceptable results. 2. Using Index 0 systems gives superior results to those with Index systems. 3. Using all available invariants for the system is necessary, including total energy.

20 Problem Constraints Not Using Total Energy. Index constraints, M=2: ( ) ( ) 2, 2 y y L + G t y = yy 3+ yy Index 0 constraints, M=3: 2 G ( t, y) = y y + y y ( y y2 L ) yl 5 + y3 + y4 gy2

21 Index_ Index_0 Solution

22 Problem Constraints Using Total Energy. Index constraints, M=3: ( y + y2 L ) 2 ˆ G ( t, y) = y y + y y ( y3 + y4) + y2g 2. Index 0 constraints, M=4: ( y + y2 L ) ( t y) Gˆ, y5l + y3 + y4 gy2 yy + yy = 2 2 ( y3 + y4) + y2g 2

23 Index_ Index_0 Solution

24 Software Issues A code, DASSLX, was developed as a modified version of DASSL but with newly added features:. Constraints are allowed and maintained 2. Step-size smoothing is implemented using a method due to G. Söderlind. 3. An extended interface allows for userdefined linear solvers and either reverse or forward communication.

25 Software Issues, Negatives The preferred starting algorithm has been replaced by Petzold. This work is called daspk3.. It is found in later DASPK codes. DASSLX uses the starting algorithm found in the older versions of DASSL. There are occasional convergence failures. The DASSLX code is complex and hard to maintain.

26 Last Slide Questions?