CL 701: Computational/Numerical Methods . . . Autumn 2009-10 CDEEP - - PowerPoint PPT Presentation

CL 701: Computational/Numerical Methods . . . Autumn 2009-10 Department: Chemical Engineering Instructor: Prof. Santosh K. Gupta E-Mail ID: sk.gupta@che.iitb.ac.in Website Address: http://www.che.iitb.ac.in/faculty/skg.html

Slide No: 24-a Course Name Lecture No. 24-a Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009)

• Prof. Santosh K. Gupta

CDEEP IIT Bombay

Date: 05 Oct 2009 Lecture No.: 24 Lecture Name: STABILITY OF ODE-IVP ALGORITHMS -- 2 Review: STABILITY OF ODE-IVP ALGORITHMS -- I Sub-Topics: STABILITY OF ODE-IVP ALGORITHMS -- 2

CDEEP IIT Bombay

Course Name Lecture No. 24-b Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009)

• Prof. Santosh K. Gupta

Slide No: 24-b

STABILITY: EXPLICIT EULER

• CHARACTERISTIC EQUATION: –

~; NOTE: NO SPURIOUS ROOT NO QUESTION OF ‘JUNK’ COMING OUT DUE TO SPURIOUS ROOTS (OF THE CHARACTERSTIC EQUATION). BUT SOMETHING ELSE HAPPENS

Slide No: 23-i/24-c Course Name Lecture No. 23-i/24-c Instructor’s Name CL 701 Computational Methods in Chemical Engineering (01/05 Oct 2009)

• Prof. Santosh K. Gupta

CDEEP IIT Bombay

CDEEP IIT Bombay

Slide No: 23-j/24-d Course Name Lecture No.23-j/24-d Instructor’s Name CL 701 Computational Methods in Chemical Engineering (01 Oct 09/05 Oct 2009)

• Prof. Santosh K. Gupta

INSTABILITY DUE TO DIFFERENT REASONS: MID-POINT RULE SPURIOUS ROOTS EXPLICIT EULER PROPAGATION OF ROUND OFF ERRORS

IMPLICIT EULER (1st ORDER ADAMS MOULTON) ;

• ; I.C.

CHARACTERSTIC EQUATION:

• ??

(YES; PADE) 2nd ORDER ADAMS-MOULTON (CRANK-NICHOLSON)

• (PADE)

CDEEP IIT Bombay

Slide No: 24-e Course Name Lecture No24-e Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009)

• Prof. Santosh K. Gupta

• ; ,

ANALYTICAL SOLUTION: NUMERICAL SOLUTION: … , IF WE WANT SMALL: ~ ~

CDEEP IIT Bombay

Slide No: 24-f Course Name Lecture No. 24-f Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009)

• Prof. Santosh K. Gupta

~ || 1 2nd REQUIREMENT OF STABILTY (ELSE ∞ n ∞) EXPLICIT EULER: HAVE A SINGLE (GENUINE) ROOT: STABILITY: || 1 | | 1 REAL : EXPLICIT EULER STABLE TILL ||

CDEEP IIT Bombay

Slide No: 24-g Course Name Lecture No. 24-g Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009)

• Prof. Santosh K. Gupta

EXPLICIT EULER: IMPLICIT EULER:

• CN

:

• EXAMPLE: FOR EXPLICIT EULER: ;
• FOR STABILITY

(SEE ‘BLOWING UP’ FOR h = 0.8 ON LECTURE NO. 24 d)

CDEEP IIT Bombay

Slide No: 24-h Course Name Lecture No. 24-h Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009)

• Prof. Santosh K. Gupta

EXPLICIT EULER: || | | COULD BE COMPLEX: INHERENTLY STABLE: Re (h) < 0 | | CIRCLE: CENTER AT z = -1, RADIUS = 1 z: COMPLEX VARIABLE

CDEEP IIT Bombay

Slide No: 24-i Course Name Lecture No. 24-i Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009)

• Prof. Santosh K. Gupta

Re (hλ) vs. Im (hλ)