Computational Methods in Engineering: ODE Solutions
Explore ordinary differential equations (ODE) and computational methods in engineering with a focus on multi-step, BDF, and Runge-Kutta techniques. Learn about initial value problems, explicit and implicit methods, as well as applications to systems of ODEs. The content covers multi-step methods such as Euler Forward, Adams-Bashforth, Euler Backward, Trapezoidal, Runge-Kutta methods, and more. Gain insights into consistency, stability, and convergence in solving ODEs.
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ESO 208A: Computational Methods in Engineering Ordinary Differential Equation: Multi- Step, BDF, Runge-Kutta Abhas Singh Department of Civil Engineering IIT Kanpur Acknowledgements: Profs. Saumyen Guha and Shivam Tripathi (CE)
ODE: Introduction We will consider general problems of the form: ?? ??= ? ?,? Solution of this equation is a function y(t) Starting from ?0, we shall take discrete time steps ?1,?2, ?? of size h such that, ??= ?0+ ? Starting from the known initial value ?0, we shall compute values of y at each time step, ?1,?2, ?? , i.e., compute tab(y) An obvious way can be: + 2 2!?? Neglecting, h2 and higher order terms: ??+1 ??+1= ??+ ?? ? ?0 = ?0 ? 0 + 3 + 4 ??+ 5 ?+ 6 ?? ??+1= ??+ ?? 3!?? 4!?? 5!?? 6!?? ??+1= ??+ ? ??,??
ODE: Introduction ??+1 ?? ??+1= ??+ ? ??,?? = ? ??,?? The method is equivalent to making a forward difference approximation of dy/dt at the nth node. It is known as the Euler Forward Method. Why not make a backward difference approximation of dy/dt at the nth node? ?? ?? 1 This is known as the Euler Backward Method. Instead of evaluating the function f either at the nth node or at the (n + 1)th node, if we take the average of the two: ??+1= ??+ 2? ??+1,??+1 + ? ??,?? = ? ??,?? ??+1= ??+ ? ??+1,??+1
ODE: Introduction ??+1= ??+ 2? ??+1,??+1 + ? ??,?? This method may also be seen as follows: ??+1 ??+1 ?? ??= ? ?,? ?? = ? ?,? ?? ?? ?? Left side integral is straight forward. Use Trapezoidal Method for the right side integral. ??+1 ??= 2? ??+1,??+1 + ? ??,?? This is known as the Trapezoidal Method.
ODE: Introduction Blue: Euler Forward, Green: Euler Backward, Red: Trapezoidal Explicit vs. Implicit methods! ??+1= ??+ ? ??,?? ??+1= ??+ ? ??+1,??+1
Ordinary Differential Equation The methods for Initial Value Problems (IVPs): Multi-step Methods Explicit: Euler Forward, Adams-Bashforth Implicit: Euler Backward, Trapezoidal and Adams-Moulton Backward Difference Formulae (BDF) Runge-Kutta Methods Applications, Startup, Combination Methods (Predictor- Corrector) Consistency, Stability, Convergence Application to System of ODEs Boundary Value Problems (BVPs) Shooting Method Direct Methods
, Multi-Step Methods: Explicit General form of multi-step or Adams-Bashforth methods: ? ??+1 ??+1= ??+ ???? ? ?=0 k= 0, 1, 2, ., n and h is the uniform time step size Example: k = 3 ??+1= ??+ ?0??+ ?1?? 1+ ?2?? 2 ?? ??= ? = ? ??+1= ??+ ?0? ?+ ?1? ? 1+ ?2? ? 2 Let s expand all the terms in Taylor s series and equate LHS with RHS!
, Multi-Step Methods: Explicit ??+1= ??+ ?0? ?+ ?1? ? 1+ ?2? ? 2 Expanding all the terms in Taylor s series: ?+ 2 ?+ 3 ?+ 4 ??? = ??+1= ??+ ? 2!? 3!? ??+ ? 5 4!?? Now, ?+ 2 ? 3 2 2 2! ? ? 1= ? ? ? 2!? ??+ ? 4 2 3 3! 3!?? ? ? 2= ? ? 2 ? ? ??+ ? 4 ?+ ? ?? Put these in the original equation! RHS = ??+1 ?+ 2 ? 3 = ??+ ?0? ?+ ?1 ? ? ? 2!? ??+ ? 4 3!?? 2 2 2! 2 3 3! + ?2 ? ? 2 ? ? ??+ ? 4 ?+ ? ??
, Multi-Step Methods: Explicit Thus, we have reduced ??+1= ??+ ?0? ?+ ?1? ? 1+ ?2? ? 2 to: ?+ 2 ?+ 3 ?+ 4 ??+ ? 2!? 3!? ??+ ? 5= ??+ ?0? 4!?? ?+ ?+ 2 ? 3 ?1 ? ? ? 2!? ??+ ? 4 3!?? + 2 2 2! 2 3 3! ?2 ? ? 2 ? ? ??+ ? 4 ?+ ? ?? Grouping Terms: ?+ 2 ?+ 3 ?+ 4 ??+ ? 2!? 3!? ?+ 3?1 ??+ ? 5= ??+ ?0+ ?1+ ?2? 4!?? 2+ 2?2 ? + 4 ?1 ?+ 6 4?2 2 ?1 2?2? ??+ ? 5 ?? 3
, Multi-Step Methods: Explicit ??+1= ??+ ?0??+ ?1?? 1+ ?2?? 2 ?+ 2 ?+ 3 ?+ 4 ??+ ? 2!? 3!? ?+ 3?1 ??+ ? 5= ??+ ?0+ ?1+ ?2? 4!?? 2+ 2?2 ? + 4 ?1 ?+ 6 4?2 2 ?1 2?2? ??+ ? 5 ?? 3 Equating both sides: ?1+ 2?2= 1 12,?1= 4 ?1 2+ 2?2=1 ?0+ ?1+ ?2= 1; 2; 6 ?0=23 5 12 3 and ?2= 23 12?? 4 5 12?? 2 ??+1= ??+ 3?? 1+
, Multi-Step Methods: Explicit Effective approximation is: 23 12?? 4 5 12?? 2 ??+1 ??+1= ??+ 3?? 1+ ?0=23 12,?1= 4 5 12 (1) 3 and ?2= We already have: ?+ 2 ?+ 3 ?+ 4 ??+ ? 5 (2) ??+1= ??+ ? 2!? 3!? ?+ 2 ?1 2?2? 4!?? ?+ 3?1 (3) ??+1= ??+ ?0+ ?1+ ?2? + 4 ?1 3 ?+ 2 2+ 2?2 ? 6 4?2 ??+ ? 5 ?? ?+ 3 ?+ 2 ? 4 ?+ 3 = ??+ ? 2? 6? ??+ ? 5 3?? ?= ??+1+ 4 ??+ ? 2? 6? ??+ ? 5 3??
, Multi-Step Methods: Explicit Thus, we have shown that effective approximation is: 23 12?? 4 ?+ 3 ?+ 3 6? ??+ ? 5= ??+1+3 4 12?? 2 +3 4 8 5 12?? 2 ??+1 ??+1= ??+ 3?? 1+ ?+ 4 ?= ??+1+ 4 ?+ 2 ?+ 2 ???,????? ??+1= ??+ ? 2!? 3!? ??+ ? 5 4!?? ???? ???? ?????:??+ ? ??+1= ??+1+ 4 2? ??+ ? 5 3?? ??+ ? 5+ 4 ??+ ? 5 3?? 23 12?? 4 4!?? 5 ?? 8 ??+ ? 5 ??+1= ??+ 3?? 1+ ?? Local truncation error (LTE) of this method is O(h4)! The method is non-self starting, or cannot be started with the given initial condition y = y0 at t = t0 or 0. Why???
, Multi-Step Methods: Explicit 5 12?? 2 +3 4 23 12?? 4 ??+ ??? ??+1= ??+ 3?? 1+ ?? 8 Let us assume that we have obtained y1 at t1 = t0 + h and y2 at t2 = t0 + 2h using another method and then applying this method for subsequent time steps: 23 12?2 4 23 12?3 4 23 12?2+ ?3 4 + ??? 5 12?0 +3 4 5 12?1 +3 4 ??+ ??? ?3= ?2+ 3?1+ ?2 8 ??+ ??? ?4= ?3+ 3?2+ ?3 8 5 12?0+ ?1 +3 4 ??+ ?3 ?? = ?2+ 3?1+ ?2 + ?2 8 This way, if we apply the method for n time steps, ?+1 ?? 4 ? ? 1 ?+1 ?? +3 4 23 12 ?=2 5 12 ?=0 ??+ ??? ??+2= ?2+ 3 ?=1 ??+ ?? 8 ?=2
, Multi-Step Methods: Explicit ?+1 ? ? 1 ?+1 ?? +3 4 23 12 ?=2 ?? 4 5 12 ?=0 ??+ ??? ??+2= ?2+ 3 ?=1 ??+ ?? 8 ?=2 Applying the first mean value theorem for integrals: ?+1 ????? = ????? =??+1 ?2 ???? ; ? ??+1,?2 ?=2 Therefore, ??+2 ?+1 ? ? 1 ?? +3 3 23 12 ?=2 ?? 4 5 12 ?=0 ??+1 ?2???? + ??? = ?2+ 3 ?=1 ??+ 8 Global truncation error (GTE) of this method is O(h3)! A method is always referred to with it s order of accuracy of GTE! Therefore, this is 3rd order Adams-Bashforth method!
, Multi-Step Methods: Explicit ? ??+1= ??+ ???? ? ?=0 Some commonly used explicit methods: GTE Order Name k Method Euler Forward 0 h ??+1= ??+ ?? 3 2?? 1 1 h2 ??+1= ??+ 2?? 1 Adams- Bashforth 23 12?? 4 5 2 h3 ??+1= ??+ 3?? 1+ 12?? 2 55 24?? 59 24?? 1+37 24?? 2 3 3 h4 ??+1= ??+ 8?? 3
, Multi-Step Methods: Implicit General form of multi-step implicit methods: ? ??+1 ??+1= ??+ ????+1 ? ?=0 k= 0, 1, 2, ., (n + 1) and h is the uniform time step size ??+1= ??+ ?0??+1+ ?1??+ ?2?? 1 Example: k = 2 ??+1= ??+ ?0??+1+ ?1??+ ?2?? 1 ?? ??= ? = ? + ?2?? 1 ??+1= ??+ ?0??+1 + ?1?? Let s expand all the terms in Taylor s series and equate LHS with RHS!
, Multi-Step Methods: Implicit + ?2?? 1 ??+1= ??+ ?0??+1 + ?1?? Expanding all the terms in Taylor s series: + 2 + 3 + 4 ??+ ? 5 ??? = ??+1= ??+ ?? 2!?? 3!?? 4!?? + 2 3 ?? ??+ ? 4 ?? 1 = ?? 2!?? 3!?? + 2 + 3 + ?? ??+ ? 4 ??+1 = ?? 2!?? 3!?? Put these in the original equation! RHS = ??+1 ??+1 + 2 + 3 + ?? ??+ ? 4 = ??+ ?0 ?? 2!?? 3!?? + ?1?? + 2 3 ?? ??+ ? 4 + ?2 ?? 2!?? 3!??
, Multi-Step Methods: Implicit + 2 + 3 + 4 ??+ ? 5 ??? = ??+1= ??+ ?? RHS = ??+1 ??+1 2!?? 3!?? 4!?? + 2 + 3 + ?? ??+ ? 4 = ??+ ?0 ?? 2!?? 3!?? + ?1?? + 2 3 ?? ??+ ? 4 + ?2 ?? 2!?? 3!?? + 3?0 2+?2 = ??+ ?0+ ?1+ ?2? ?+ 2?0 ?2?? ?? 2 + 4?0 6 ?2 ??+ ? 5 ?? 6 Comparing two sides: ?0 ?2=1 ?0+ ?2=1 ?0+ ?1+ ?2= 1; 2; 3
, Multi-Step Methods: Implicit ?0 ?2=1 ?2= 1 ?0+ ?2=1 ?0+ ?1+ ?2= 1; 5 12, The method is: 2; 3 ?1=2 ?0= 3, 12 5 12??+1+2 + 2 2!?? ?+ 2 2?? ??+ 4 4!?? 1 12?? 1 ??+1= ??+ 3?? + 4 + 3 ??+ ? 5 LHS = ??+1= ??+ ?? 3!?? 4!?? + 3 + 4 RHS = ??+1= ??+ ? ??+1= ??+1 4 ??+ ? 5 6?? 12?? ??+ ? 5= ??+1 4 ??? 5 12?? 24?? The LTE of the method is O(h4) and GTE is O(h3). This is the 3rd order Adams-Moulton method!
, Multi-Step Methods: Implicit ? ??+1 ??+1= ??+ ????+1 ? ?=0 Some commonly used implicit methods: GTE Order Name k Method Euler Backward 0 h ??+1= ??+ ??+1 1 2??+1+1 1 h2 Trapezoidal ??+1= ??+ 2?? 12??+1+2 5 1 2 h3 ??+1= ??+ 3?? 12?? 1 Adams- Moulton 3 8??+1+19 5 1 3 h4 ??+1= ??+ 24?? 24?? 1+ 24?? 2
, Backward Difference Formulae (BDF) Explicit multi-step methods: ? ?? ??= ? ?,? ??+1 ?? = ???? ? ?=0 k= 0, 1, 2, ., n and h is the uniform time step size Implicit multi-step methods: ? ?? ??= ? ?,? ??+1 ?? = ????+1 ? ?=0 k= 0, 1, 2, ., (n + 1) and h is the uniform time step size All variations were in the evaluations of f. What happens if we keep the f evaluation at only one point and use multi-point approximation of the derivative dy/dt?
, Backward Difference Formulae (BDF) Backward Difference Formulae or BDFs: ? ????+1 ?= ??+1 ?=0 k= 0, 1, 2, ., (n + 1) and h is the uniform time step size Example: k = 2 ?0??+1+ ?1??+ ?2??+1= ??+1 ?? ??= ? = ? Let s expand all the terms in Taylor s series and equate LHS with RHS! ?0??+1+ ?1??+ ?2?? 1= ??+1
, Backward Difference Formulae (BDF) ?0??+1+ ?1??+ ?2?? 1= ??+1 + 2 2!?? + 2 2!?? + 2 2!?? + 3 + 4 ??+ ? 5 ??+1= ??+ ?? 3!?? 4!?? 3 + 4 ??+ ? 5 ?? 1= ?? ?? 3!?? 4!?? + 3 + ?? ??+ ? 4 ??+1 ??? = = ?? 3!?? + 2 = ?0+ ?1+ ?2??+ ?0 ?2?? + 3 3! ?0+ ?2?? 2! + 4 ??+ ? 5 ?0 ?2?? ?0+ ?2?? + 4 4! + 3 + 2?? ??+ ? 5 ??? = ?? 2!?? 3!?? ?0+ ?1+ ?2= 0 ?0 ?2= 1 ?0+ ?2= 2
, Backward Difference Formulae (BDF) ?0??+1+ ?1??+ ?2?? 1= ??+1 ?0+ ?1+ ?2= 0 ?0 ?2= 1 ?0+ ?2= 2 ?0=3 ?2=1 2 ?1= 2 2 3??+1 4??+ ?? 1= 2 ??+1 Comparing LHS and RHS, the truncation error term is: ?? = 4 4! The LTE of the method is O(h4) and the GTE is O(h3). ??+ 4 ??+ ? 5= 4 ??+ ? 5 ?0+ ?2?? 3!?? 12?? This is the 3rd order BDF!
