The Computable Differential Equation Lecture ... - Bruce E. Shapiro

180 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS
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Algorithm 9.2. Semi-explicit Index-1 DAE with Backward Euler
To solve the semi-implicit index-1 DAE
y ′ = f(t, y, z) (9.148)
0 = g(t, y, z) (9.149)
where g z is non-singular, at each time step t n solve the constraint for z n
and substitute the result back into the ODE,
z = h(t n , y n ) (9.150)
y ′ n = f(t n , y n , h(t n , y n )) (9.151)
(9.152)
This produces an ordinary differential system that can be discretized using backwards-
Euler (or any other method of choice) via
1
h (y n − y n−1 ) = f(t n , y n , h(t n , y n )) (9.153)
Solve for y n using a Newton root-finder, and then proceed to the next mesh point.
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Let us proceed by implementing an equation-specific solution for the implicit DAE
A ′ = −αAE + βX (9.154)
B ′ = γX (9.155)
X ′ = αAE − (γ + β)X (9.156)
0 = E + X − 1 (9.157)
Using the backward Euler method, the iteration formulas are
0 = F 1 (A n , B n , X n , E n ) = 1 h (A n − A n−1 ) + αA n E n − βX n (9.158)
0 = F 2 (A n , B n , X n , E n ) = 1 h (B n − B n−1 ) − γX n (9.159)
0 = F 3 (A n , B n , X n , E n ) = 1 h (X n − X n−1 ) − αA n E n + (γ + β)X n (9.160)
0 = F 4 (A n , B n , X n , E n ) = E n + X n − 1 (9.161)
We need to solve this explicitly at each time-step for the values of A n , B n , E n , X n ;
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007