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Numerical Methods for Ordinary Differential Equation with
Initial Condition:
d y / d t = f ( t , y ) with
y( t = a ) = y_0, a <= t <= b
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General Questions
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We have existence and uniqueness of the solution when f(t,y)
is continuous and Lipschitz w.r.t its second variable. |
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Higher order ODE can always been put in the form above. |
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local truncation error, global error |
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well-posed, stability, A-stability, consistency, convergence |
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ill-conditioned |
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stiffness (prefer to use implicit schemes) |
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Numerical Schemes
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Explicit Schemes
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Euler's Method (first order) |
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Higher Order Taylor Methods
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give higher order schemes by
calculating higher derivatives of f |
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Adams-Bashforth Methods
(multi-step)
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give higher order schemes by
profiting from history |
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need to use one-step method (at
most one order lower) to give starting values |
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ex: AB2 |
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Explicit Runge-Kutta Methods (one-step)
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give higher order schemes by using
more evaluations of f |
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ex: Modified Euler, Midpoint
Method, Heun's Method, RK4 |
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Implicit Schemes
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Backward Euler's Method (first
order) |
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Implicit Runge-Kutta Methods (one-step) |
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Adams-Moulton Methods (multi-step)
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give higher order schemes by
profiting from history |
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Note: need to solve the equation
for y_(k+1) algebraically, iterative method or use predictor-corrector approach
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