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

Slide No: 24-a CL 701: Computational/Numerical Methods . . . Autumn 2009-10 CDEEP IIT Bombay 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 Course Name Lecture No. 24-a Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009) Prof. Santosh K. Gupta

Slide No: 24-b Date: 05 Oct 2009 CDEEP Lecture No.: 24 IIT Bombay Lecture Name: STABILITY OF ODE-IVP ALGORITHMS -- 2 Review: STABILITY OF ODE-IVP ALGORITHMS -- I Sub-Topics: STABILITY OF ODE-IVP ALGORITHMS -- 2 Course Name Lecture No. 24-b Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009) Prof. Santosh K. Gupta

STABILITY: EXPLICIT EULER Slide No: 23-i/24-c �� CDEEP �� � �� � � ��� � � � � �� �� IIT Bombay � � � �� � � � � � � �� CHARACTERISTIC EQUATION: – � � �� � ��� � � � � � � � �� �~� �� ; ��� ���� ������ NOTE: NO SPURIOUS ROOT � NO QUESTION OF ‘JUNK’ COMING OUT DUE TO SPURIOUS ROOTS (OF THE CHARACTERSTIC EQUATION). BUT SOMETHING ELSE HAPPENS 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

Slide No: 23-j/24-d CDEEP IIT Bombay INSTABILITY DUE TO DIFFERENT REASONS: MID-POINT RULE SPURIOUS ROOTS EXPLICIT EULER PROPAGATION OF ROUND OFF ERRORS 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

Slide No: 24-e IMPLICIT EULER (1 st ORDER ADAMS MOULTON) CDEEP IIT Bombay �� � ��� � � � � �� ���� ; �� � � � � �� ; I.C. CHARACTERSTIC EQUATION: � � ��� � ��� � � ?? � � � �� (YES; PADE) � � ���� 2 nd ORDER ADAMS-MOULTON (CRANK-NICHOLSON) �� �� � �� ���� � � �� � � � � � � ��� � � � � � (PADE) �� �� � Course Name Lecture No24-e Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009) Prof. Santosh K. Gupta

�� �� � �� ; � � �, � � � � Slide No: 24-f ANALYTICAL SOLUTION: � � � � � � ��� CDEEP NUMERICAL SOLUTION: � � � �� � IIT Bombay � ��� � �� ��� � ��� � � � � �� ��� � … � � ���� � � � � ���� �� � � ����� ��� �����, � � � � ��� � � � � ������� � � ���� �� � � � � � � � � �� ������� � � ���� � � � � � ���� IF WE WANT � ��� SMALL: � � ~� �� �� � � ~���� ������������� �� � �� Course Name Lecture No. 24-f Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009) Prof. Santosh K. Gupta

� � ~� �� Slide No: 24-g |� � | � 1 � 2 nd REQUIREMENT OF STABILTY (ELSE � � � ���� � ∞ �� n � ∞ ) CDEEP IIT Bombay EXPLICIT EULER: HAVE A SINGLE (GENUINE) ROOT: � � � � �� STABILITY: |�| � 1 |� � ��| � 1 � � �������� � � �������� ���������� REAL � : EXPLICIT EULER STABLE TILL |��| � � Course Name Lecture No. 24-g Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009) Prof. Santosh K. Gupta

Slide No: 24-h EXPLICIT EULER: � � � � �� CDEEP IIT Bombay � IMPLICIT EULER: � � ���� �� �� � CN : �� �� � EXAMPLE: FOR EXPLICIT EULER: � � � ���; � � � � � � � �� � � � � FOR STABILITY (SEE ‘BLOWING UP’ FOR h = 0.8 ON LECTURE NO. 24 d) Course Name Lecture No. 24-h Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009) Prof. Santosh K. Gupta

Slide No: 24-i EXPLICIT EULER: |�| � |� � ��| � � CDEEP IIT Bombay � COULD BE COMPLEX: INHERENTLY STABLE: Re (h) < 0 Re (h λ ) vs . Im (h λ ) |� � �| � � � CIRCLE: CENTER AT z = -1, RADIUS = 1 z: COMPLEX VARIABLE Course Name Lecture No. 24-i Instructor’s Name CL 701 Computational Methods in Chemical Engineering (05 Oct 2009) Prof. Santosh K. Gupta