An Implicit One-Step Method
نویسنده
چکیده
We have learned that the numerical solution obtained from Euler's method, converges to the exact solution () of the initial value problem ′ = (,), (0) = 0 , as ℎ → 0. We now analyze the convergence of a general one-step method of the form +1 = + ℎΦ(, , ℎ), for some continuous function Φ(, , ℎ). We define the local truncation error of this one-step method by (ℎ) = (+1) − () ℎ − Φ(, (), ℎ). That is, the local truncation error is the result of substituting the exact solution into the approximation of the ODE by the numerical method. We therefore say that the one-step method is consistent if A consistent one-step method is one that converges to the ODE as ℎ → 0. We then say that a one-step method is stable if Φ(, , ℎ) is Lipschitz continuous in. That is,
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