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The relation between local and global truncation errors is slightly different from in the simpler setting of one-step methods. For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors.
(Figure 2) Illustration of numerical integration for the equation ′ =, = Blue is the Euler method; green, the midpoint method; red, the exact solution, =. The step size is =
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The step size is =. The same illustration for = The midpoint method converges faster than the Euler method, as .. Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs).
In general, a method with (+) LTE (local truncation error) is said to be of kth order. The region of absolute stability for the backward Euler method is the complement in the complex plane of the disk with radius 1 centered at 1, depicted in the figure. [ 4 ]
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Lady Windermere's Fan (mathematics) — telescopic identity relating local and global truncation errors Stiff equation — roughly, an ODE for which unstable methods need a very short step size, but stable methods do not