1. CSE245: Computer-Aided Circuit Simulation and Verification Lecture Note 5 Numerical Integration Prof. Chung-Kuan Cheng 1

2. Numerical Integration: Outline One-step Method for ODE (IVP) –Forward Euler –Backward Euler –Trapezoidal Rule –Equivalent Circuit Model Convergence Analysis Linear Multi-Step Method Time Step Control 2

3. Ordinary Difference Equations N equations, n x variables, n dx/dt. Typically analytic solutions are not available  solve it numerically 3

4. Numerical Integration Forward Euler Backward Euler Trapezoidal 4

5. Numerical Integration: State Equation Forward Euler Backward Euler 5

6. Numerical Integration: State Equation Trapezoidal 6

7. 7 Equivalent Circuit Model-BE Capacitor + C - + - + -

8. 8 Equivalent Circuit Model-BE Inductor + L - + - + -

9. 9 Equivalent Circuit Model-TR Capacitor + C - + - + -

10. 10 Equivalent Circuit Model-TR Inductor + L - + - + -

11. Trap Rule, Forward-Euler, Backward-Euler All are one-step methods x k+1 is computed using only x k, not x k-1, x k-2, x k-3... Forward-Euler is the simplest No equation solution explicit method. Backward-Euler is more expensive Equation solution each step implicit method most stable (FE/BE/TR) Trapezoidal Rule might be more accurate Equation solution each step implicit method More accurate but less stable, may cause oscillation Summary of Basic Concepts 11

12. 12 Stabilities Froward Euler

13. Difference Eqn Stability region 1 Im(z) Re(z) Forward Euler ODE stability region Region of Absolute Stability FE region of absolute stability 13

14. 14 Stabilities Backward Euler

15. Difference Eqn Stability region 1 Im(z) Re(z) Backward Euler Region of Absolute Stability BE region of absolute stability 15

16. 16 Stabilities Trapezoidal

17. Convergence Consistency: A method of order p (p>1) is consistent if Stability: A method is stable if: Convergence: A method is convergent if: Consistency + Stability Convergence 17

18. A-Stable Dahlquist Theorem: –An A-Stable LMS (Linear Multi-Step) method cannot exceed 2nd order accuracy The most accurate A-Stable method (smallest truncation error) is trapezoidal method. 18

19. Convergence Analysis: Truncation Error Local Truncation Error (LTE): –At time point t k+1 assume x k is exact, the difference between the approximated solution x k+1 and exact solution x * k+1 is called local truncation error. –Indicates consistency –Used to estimate next time step size in SPICE Global Truncation Error (GTE): –At time point t k+1, assume only the initial condition x 0 at time t 0 is correct, the difference between the approximated solution x k+1 and the exact solution x * k+1 is called global truncation error. –Indicates stability 19

20. LTE Estimation: SPICE Taylor Expansion of x n+1 about the time point t n : x(t n+1 )=x(t n )+dx(t n )/dt∙h+d 2 x(t n )/dt 2 ∙h 2 /2!+d 3 x(t n )/dt 3 ∙h 3 /3!+… Taylor Expansion of x n about the time point t n+1 : x(t n )=x(t n+1 )-dx(t n+1 )/dt∙h+d 2 x(t n+1 )/dt 2 ∙h 2 /2!-d 3 x(t n+1 )/dt 3 ∙h 3 /3!+… Forward Euler –Exercise x(t n+1 )-x n+1 =x(t n+1 )-(x n +hdx n /dt) Backward Euler –Exercise x(t n+1 )-x n+1 =x(t n+1 )-(x n +hdx n+1 /dt) Trapezoidal –Exercise x(t n+1 )-x n+1 =x(t n+1 )-[x n +(dx n /dt+dx n+1 /dt)h/2] LTE 20

21. Formula for pth order method LTE 21 Formula E(x(t),h)=∑ i=0,k a i x(t n-i )+hb i dx(t n-i )/dt Let x(t)=[(t n -t)/h] p E(x(t),h)=∑ i=0,k a i [(t n -t n-i )/h] p -pb i [(t n -t n-i )/h] p-1 =∑ i=0,k a i i p -pb i i p-1 If the formula is a pth order method, we have Case p=0: ∑ i=0,k a i =0 Case p=1: ∑ i=0,k a i i-b i =0 …… Case p: ∑ i=0,k (a i i-pb i )i p-1 =0

22. Formula for pth order method: Example LTE 22 Forward Euler: We have a 0 =1, a 1 =-1, b 0 =0, b 1 =-1 Case p=0: ∑ i=0,k a i =0 Case p=1: ∑ i=0,k a i i-b i =0 Case p=2: ∑ i=0,k (a i i-2b i )i=-1+2=1 Backward Euler: We have a 0 =1, a 1 =-1, b 0 =-1, b 1 =0 Case p=0: ∑ i=0,k a i =0 Case p=1: ∑ i=0,k a i i-b i =0 Case p=2: ∑ i=0,k (a i i-2b i )i=-1+2=1 Trapezoidal Rule: We have a 0 =1, a 1 =-1, b 0 =-1/2, b 1 =-1/2 Case p=0: ∑ i=0,k a i =0 Case p=1: ∑ i=0,k a i i-b i =0 Case p=2: ∑ i=0,k (a i i-2b i )i=0 Case p=3: ∑ i=0,k (a i i-3b i )i 2 =-1+3/2=1/2

23. Formula for pth order method: Variables LTE 23 There are 2(k+1)-1 unknowns (a 0 =1), and p+1 equations. Thus, we need 2(k+1)-1 ≥ p+1 In other words, k ≥ p/2

24. Formula for pth order method: Local Truncation Error LTE 24 By Taylor’s expansion, we have x(t)=x(t n )+x’(t n )(t-t n )+…+1/(p+1)! x (p+1) (t n )(t-t n ) p+1 +… Thus, the error of pth order method is ∑ i=0,k {a i [(t n -t n-i )/h] p+1 -pb i [(t n -t n-i )/h] p }1/(p+1)! x (p+1) (t n )h p+1 +O(h p+2 ) Let us set E p+1 ={∑ i=0,k a i i p+1 -pb i i p }/p! Methoda0a0 a1a1 b0b0 b1b1 E p+1 FE10 E 2 =1/2 BE1 0E 2 =1/2 TR1-1/2 E 3 =1/12

25. Time Step Control: SPICE We have derived the local truncation error the unit is charge for capacitor and flux for inductor Similarly, we can derive the local truncation error in terms of (1) the unit is current for capacitor and voltage for inductor Suppose E D represents the absolute value of error that is allowed per time point. That is together with (1) we can calculate the time step as 25

26. Time Step Control: SPICE (cont ’ d) DD 3 (t n+1 ) is called 3rd divided difference, which is given by the recursive formula 26

27. Multiple Step Integration: Stability For a system x’=λx, let q= λh. The integration formula is ∑ i=0,k a i x n-i +hb i x’ n-i =0. We set (a 0 +qb 0 )z k + (a 1 +qb 1 )z k-1 …+(a k +qb k )=0 There are k roots, r i, of the polynomial eq. The generic solution is x n =c 1 r 1 n +c 2 r 2 n +…+c k r k n but for multiplicity root, we have x n =…+(c i0 +c i1 n+…+c im n m-1 )r i n +… If |r i |< 1 for all i, the system is stable Else for |r i |= 1 but not multiplicity root, the system remains stable 27

28. Multiple Step Integration: Stability For a system x’=λx, let q= λh. The integration formula is ∑ i=0,k a i x n-i +hb i x’ n-i =0. We set (a 0 +qb 0 )z k + (a 1 +qb 1 )z k-1 …+(a k +qb k )=0 Examples using FE, BE, TZ methods 28 Methoda0a0 a1a1 b0b0 b1b1 root FE10 1+q BE1 01/(1-q) TR1-1/2 (1+q/2)/(1-q/2)

29. Multiple Step Integration: Stability For a system x’=λx, let q= λh. The integration formula is ∑ i=0,k a i x n-i +hb i x n-i =0. A Stability: The system is stable for all Real(q)≤ 0. Dahlquist’s barrier: –An A-Stable LMS (Linear Multi-Step) method cannot exceed 2nd order accuracy 29