Practical Tests with Global Error Evaluation Techniques in Multistep Methods

A. Ramanantoanina, Dr G. Kulikov

SANUM2006

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ODE’s & Multistep Methods

Consider the system of ODE

x′(t) = f(x(t)), t ∈ [0, T], x(t_{0}) = x_{0}, (1)

and the grid

w = {t_{0} = 0, t_{1}, t_{2}, …, t_{N} = T } Multistep methods

∑_{j=0}^{l} α_{j}x_{k+1−j} = h ∑_{j=0}^{l} β_{j}f(x_{k+1−j})  (2)

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Local & Global Errors

Local error

e(t_{k}) = x(t_{k}) − x̃_{k}

where x̃_{k} is computed using a multistep method with exact input.

Global error

E(t_{k}) = x(t_{k}) − x_{k}

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Global error evaluation

Methods:

Solving for the correction

E′_{h}(t) = f(P_{h} − E_{h}(t)) − P′_{h}(t), E_{h}(0) = 0 Zadunaisky’s technique

x̃′ = f(x̃) + d(h), x̃(0) = x(0) where the perturbation d(h) = P′_{h} − f(P_{h}) Solving the linearised discrete variational equation

E_{k} = P_{k}C

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Global error evaluation

Methods:

Solving for the correction

E′_{h}(t) = f(P_{h} − E_{h}(t)) − P′_{h}(t), E_{h}(0) = 0 Zadunaisky’s technique

x̃′ = f(x̃) + d(h), x̃(0) = x(0) where the perturbation d(h) = P′_{h} − f(P_{h}) Solving the linearised discrete variational equation

E_{k} = P_{k}C

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Global error evaluation

Methods:

Solving for the correction

E′_{h}(t) = f(P_{h} − E_{h}(t)) − P′_{h}(t), E_{h}(0) = 0 Zadunaisky’s technique

x̃′ = f(x̃) + d(h), x̃(0) = x(0) where the perturbation d(h) = P′_{h} − f(P_{h}) Solving the linearised discrete variational equation

E_{k} = P_{k}C

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Global error evaluation (Cont)

Richardson extrapolation

E_{k} = \frac{x_{k}^{h} − xk^{\frac{h}{2}}}{1 − 2^{−s}}

where s is the order of the multistep method. Using two different methods

E_{k} = x_{k}^{1} − x_{k}^{2}

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Global error evaluation (Cont)

Richardson extrapolation

E_{k} = \frac{x_{k}^{h} − xk^{\frac{h}{2}}}{1 − 2^{−s}}

where s is the order of the multistep method. Using two different methods

E_{k} = x_{k}^{1} − x_{k}^{2}

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Test Problems

An oscillatory problem described by

x′(t) = x(t)cos(t),

with the initial condition x(0) = 1 for t ∈ [0, 1]. The exact solution to this problem is

x(t) = e^{sin(x(t))}.

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Test Problems (Cont)

A stiff ODE: Butcher test problem

x′(t) = λ(x(t) − sin(μt)) + μ cos(μt) for t ∈ [0, 1] with the initial condition x(0) = 1. It is known that the solution to this problem is:

x(t) = sin(μt) + e^{λt}.

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Graphs

We present in the following figures the graphs of the errors in the estimation provided by different

methods.

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Conclusion

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