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MATH 175: NUMERICAL ANALYSIS II

MATH 175: NUMERICAL ANALYSIS II. Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB. VARIABLE STEP-SIZE METHODS (ADAPTIVE). It is a good practice to change the step size for a solution that moves between periods of slow change and periods of fast change.

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MATH 175: NUMERICAL ANALYSIS II

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  1. MATH 175: NUMERICAL ANALYSIS II Lecturer: Jomar Fajardo Rabajante 2ndSemAY 2012-2013 IMSP, UPLB

  2. VARIABLE STEP-SIZE METHODS (ADAPTIVE) It is a good practice to change the step size for a solution that moves between periods of slow change and periods of fast change. One common variable step-size approach is using two solvers of different orders (embedded pairs).

  3. VARIABLE STEP-SIZE METHODS Basic idea: • The user sets a tolerance error that must be met by each step. • The method is designed to reject the step and cut the step size (usually by half) if the tolerance error is exceeded. • If the tolerance error is met too well – say the error is 1/10 of the tolerance error – then the step size is increased (usually doubled). [We will not do number 3].

  4. VARIABLE STEP-SIZE METHODS Before crudely cutting the step size into half, we will try to come-up first with a formula that might give us the right step size. We will use relative error estimate where θ>0 protects against very small values of wi.

  5. VARIABLE STEP-SIZE METHODS Let T be the tolerance (relative) error. Assume that the (explicit one-step) solver has order p, so that the local truncation error is O(hp+1). Hence, the h that satisfies (1) & (2) is

  6. VARIABLE STEP-SIZE METHODS Continuation… We consider only the 80% to be conservative (0.8 is the safety factor); and we consider theta. Hence,

  7. VARIABLE STEP-SIZE METHODS If h* does not meet the tolerance, then set hi := h* and apply again to the formula. If the newly computed h* still does not meet the tolerance, then cut h* into half (i.e. h* := h*/2). Continuously cut until the tolerance is met. If h* becomes very small (say less than 10^-10), then stop and tell the user that the step size becomes very small. However, if tolerance error is met set hi+1 := hi .

  8. VARIABLE STEP-SIZE METHODS Computing ei is a challenge, so to estimate ei , we will use a higher order method’s estimate – say zi+1 . This method is called embedded pair. Estimate from higher-order method Estimate from original method

  9. EMBEDDED RUNGE-KUTTA PAIRS 1st Example: RK 2/3 We partner explicit Trapezoid method (an RK 2 method) with an RK 3 method.

  10. EMBEDDED RUNGE-KUTTA PAIRS 1st Example: RK 2/3 Getting ei: Note: We can use zi to advance our method i.e. estimate for yi (this is local extrapolation)

  11. EMBEDDED RUNGE-KUTTA PAIRS 2nd Example: RK 4/5 or Runge-Kutta-Fehlberg (this method is famous)

  12. Continuation… 2nd Example: RK 4/5

  13. VARIABLE STEP-SIZE METHODS Other well-known methods: • Bogacki-Shampine 2/3 (ODE23 in MATLAB) • Dormand-Prince 4/5 (ODE45 in MATLAB) • RKV 5/6 (Runge-Kutta-Verner) • Bulirsch-Stoer method You can also use Richardson’s extrapolation to improve the solutions. Note: There maybe other ways of computing h* .

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