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Lecture41

# Lecture41 - Numerical Stability Amplification or decay of...

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Numerical Stability Amplification or decay of numerical errors A numerical method is stable if error incurred at one stage of the process do not tend to magnify at later stages Ill-conditioned differential equation -- numerical errors will be magnified regardless of numerical method Stiff differential equation -- require extremely small step size to achieve accurate results

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Stability Example problem e y y y 0 y ay dt dy at 0 0 - = = - = ) ( = - = - = + = + = - ε ε ε ) ( / ) ( , 0 E aE dt dE y(x) (x) y E(x) Error let e y y then y y(0) if at 0 0 = < = - stable llly exponentia decay error : 0 a stable neutrally : 0 a unstable lly exponentia grow error : 0 a e E at ε
Euler Explicit Method Stability criterion Region of absolute stability i i i i i 1 i 0 y ah 1 h ay y h dt y d y y y 0 y ay y x f dt y d ) ( ) ( ) ( ; ) , ( - = - + = + = = - = = + 1 ah 1 r o 1 y y i 1 i - + 2 ah 0 1 ah 1 1 - - Amplification factor

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Euler Implicit Method Unconditionally stable !
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• Spring '09
• RAPHAELHAFTKA
• Yi, Numerical ordinary differential equations, Stiff equation, der Pol Equation, van der Pol, Explicit and implicit methods

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Lecture41 - Numerical Stability Amplification or decay of...

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