Linear multistep method: Difference between revisions

Revision as of 06:20, 12 May 2011

Revision as of 03:38, 12 August 2011

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where ''h'' denotes the step size and ''f'' the right-hand side of the differential equation. The coefficients <math> a_0, \ldots, a_{s-1} </math> and <math> b_0, \ldots, b_s </math> determine the method. The designer of the method chooses the coefficients; often, many coefficients are zero. Typically, the designer chooses the coefficients so they will exactly interpolate <math>y(t)</math> when it is an ''n''th order polynomial.

If the value of <math> b_s</math> is= nonzero,0 then the value of <math>y_{n+s}</math> depends on the value of <math> f(t_{n+s}, y_{n+s}) </math>. Consequently,then the method is called "explicit", if <math> b_s = 0 </math>. In that case,since the formula can directly compute <math> y_{n+s} </math>. If <math> b_s \ne 0 </math> then the method is implicit and the equation for <math> y_{n+s} </math> must be solved. [[Iterative methods]] such as [[Newton's method]] are often used to solve the implicit formula.

If <math> b_s \ne 0 </math> then the method is called "implicit", since the value of <math>y_{n+s}</math> depends on the value of <math> f(t_{n+s}, y_{n+s}) </math>, and the equation must be solved for <math> y_{n+s} </math>. [[Iterative methods]] such as [[Newton's method]] are often used to solve the implicit formula.

Sometimes an explicit multistep method is used to "predict" the value of <math>y_{n+s}</math>. That value is then used in an implicit formula to "correct" the value. The result is a [[Predictor-corrector method]].