Ordinary Differential Equation The methods for Initial Value

Ordinary Differential Equation ü The methods for Initial Value Problems (IVPs): ü Multi-step Methods ü Explicit: Euler Forward, Adams-Bashforth ü Implicit: Euler Backward, Trapezoidal and Adams-Moulton ü Backward Difference Formulae (BDF) ü Runge-Kutta Methods ü Applications, Startup, Combination Methods (Predictor. Corrector) ü Consistency, Stability, Convergence ü Application to System of ODEs ü Boundary Value Problems (BVPs) ü Shooting Method ü Direct Methods

Applications: Summary of Concerns ü Accuracy of the higher order multi-step and BDF methods are affected if the starting values are used from the lower order methods. ü How to start these non-self starting algorithms? ü All implicit methods (multi-step and BDF) may involve solution of non-linear equations (if f contains a nonlinear function of the dependent variable y) ü Is there a way to avoid this solution of non-linear equations? ü Numerical oscillations (instability) observed in some methods and not in some! ü Is there a way to predict and therefore, choose correct parameters for algorithm so that the numerical oscillations can be avoided? We will do Convergence Analysis!

ODE: Consistency, Stability, Convergence (Recap; Text Book S 6. 3. 2) •

Numerical Methods for IVPs: Stability •

Stability: Model Problem •

Stability: Model Problem •

Stability: Model Problem •

Stability: Multi-Step Methods (explicit) Example •

Stability: Multi-Step Methods (explicit) Example

Stability: Multi-Step Methods (explicit) Example •

Stability: Multi-Step Methods (explicit) Example

Stability: Multi-Step (Implicit) • Euler Backward method is stable everywhere outside the circle! Homework: Stability region of the Trapezoidal Method!

Stability: Multi-Step Methods (implicit) Example •

Stability: Multi-Step Methods (implicit) Example ü Euler backward method is unconditionally stable! (It is stable everywhere, where the analytical problem is also stable) ü Trapezoidal: find as homework! ü Adams-Moulton 3 rd and 4 th order methods are conditionally stable! ü Pay attention to the stability for purely imaginary λ!

Stability: BDF Methods Example •

Stability: BDF Methods Example ü For all the BDFs: Stability Region is outside the enclosed region! ü For real λ, all the BDFs are unconditionally stable! ü One can use any h without having to worry about the stability! ü Useful for stiff equations!

Stability: Runge-Kutta Methods Example •

Stability: Runge-Kutta Methods Example ü 4 th order R-K has very good stability properties (λh up to 2. 78 on the real part and 2. 83 on the imaginary part) ü The method is also stable for purely imaginary λh ü Homework: For our problem, check the stability limits of the R-K methods!

Phase Error • For periodic functions, there are phase error associated with the numerical solution. Can we quantify them?

Phase Error •

Phase Error •

Phase Error •

Arbitrary f How to choose time step h for arbitrary nonlinear f: ü Expand f in Taylor’s series around the initial condition, ü retain the first two terms (linear terms) to obtain the equivalent model problem ü set λ = coefficient of y ü compute h from the stability diagram! ü To account for the non-linearity, stay well below the stability limit!