ECE257 Numerical Methods andECE257 Numerical Methods andScientific ComputingScientific ComputingOrdinary Differential EquationsOrdinary Differential EquationsECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutTodayToday’’s class:s class:••StiffnessStiffness••Multistep MethodsMultistep MethodsECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiff EquationsStiff Equations••Stiffness occurs in a problem where two orStiffness occurs in a problem where two ormore independent variables change at verymore independent variables change at verydifferent scalesdifferent scales••Example:Example: € dydt= −1000y + 3000 − 2000e−t y(0) = 0€ y = 3 − 0.998e−1000t− 2.002e−t••Analytical Solution:Analytical Solution:ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiff EquationsStiff Equations••As As tt moves away from zero, the solution moves away from zero, the solutionsettles tosettles to••You still need to account for the You still need to account for the ee--10001000tt term termeven after you move away from zeroeven after you move away from zero••If you donIf you don’’t do so, the system will bet do so, the system will beunstableunstable••Adaptive methods will not workAdaptive methods will not work€ y = 3 − 0.998e−1000t− 2.002e−t€ y = 3 − 2.002e−tECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiff EquationsStiff Equations€ dydt= −ay€ yi+1= yi+dyidth€ yi+1= yi− ayih€ yi+1= yi1− ah( )For the system to be stable |1-ah| must beless than 1, or h<2/a€ dydt= −1000y + 3000 − 2000e−t=> h <21000ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStabilityStability•A numerical method is stable if errors occurring at one stage of the process do not tend to be magnified at later stages.•A numerical method is unstable if errors occurring at one stage of the process tend to be magnified at later stages.•Numerical methods which may be unstable should be carefully dealt with or avoided.•In general, the stability of a numerical scheme depends on the step size. Usually, large step sizes lead to unstable solutions.•Implicit methods are in general more stable than explicit methods.ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiff EquationsStiff EquationsFrom Numerical Methods for Engineers, Chapra and Canale, Copyright © The McGraw-Hill Companies, Inc.ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiff EquationsStiff Equations1.1885141.1885141.0178631.017863h=0.001h=0.0014.346x104.346x102424-289.402-289.402h=0.01h=0.01y(0.1)y(0.1)y(0.01)y(0.01)ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiffnessStiffness••Stiffness can also cause problems evenStiffness can also cause problems evenwhen the system is stablewhen the system is stable••Example:Example: € d2ydx2= 100y y(0) = 1, y'(0) = −10 € Analytical Solution y1= e−10x, y2= −10e−10xy(0.5) = 0.00674€ dy1dx= y2dy2dx= 100y1ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiffnessStiffness0.0067380.006738h=0.00005h=0.000050.0067380.006738h=0.005h=0.0050.0067650.006765h=0.05h=0.0513.708313.7083h=0.5h=0.5y(0.5)y(0.5)ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiffnessStiffness5168265168260.0067380.006738h=0.00005h=0.000051548381548380.0067380.006738h=0.005h=0.005-158272-1582720.0067650.006765h=0.05h=0.052.34 x 102.34 x 10111113.708313.7083h=0.5h=0.5y(5)y(5)y(0.5)y(0.5)ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiffnessStiffness••Roundoff or truncation error can cause inaccuraciesRoundoff or truncation error can cause inaccuraciesin initial values and lead us to follow an inaccuratein initial values and lead us to follow an inaccuratesolutionsolution € d2ydx2= 100y y(0) = 1, y'(0) = −10€ y = Ae−10x+ Be10x••As x gets large, the As x gets large, the ee10x10x dominates and because of dominates and because ofaccumulated errors as we integrate from the initialaccumulated errors as we integrate from the initialvalues, we will introduce large errorsvalues, we will introduce large errorsECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiffnessStiffness••Implicit (backward) differencing (EulerImplicit (backward) differencing (Euler’’s)s)€ yi+1= yi+dyi+1dth€ yi+1= yi− ayi+1h€ yi+1=yi1+ ah( )••This method is unconditionally stableThis method is unconditionally stable••True for linear ODEsTrue for linear ODEsECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiffnessStiffnessFrom Numerical Methods for Engineers, Chapra and Canale, Copyright © The McGraw-Hill Companies, Inc.ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutStiffnessStiffnessFrom Numerical Methods for Engineers, Chapra and Canale, Copyright © The McGraw-Hill Companies, Inc.ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 18John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutMulti-step MethodsMulti-step Methods] ...),( ),( ),([..11211101211+++⋅+++=−−++−+iiiiiiiiiyxfbyxfbyxfbhyayay••R- R- K methods use only one previous approximation