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Lecture14

Lecture14 - LECTURE 14 Numerical Integration Numerical...

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LECTURE 14 Numerical Integration Numerical Integration proceeds from time 0 t to the next time instant h t t + = 0 where h is the step-size. For high accuracy, the step-size must be kept small. In the Runge-Kutta method of integrating ) , ( u x f dt x d = for example, the right hand-side )] ( ), ( [ t u t x f is evaluated 4 times at each step. Usually a test is made by the software regarding the accuracy of the predicted ) ( 0 h t x + . If unsatisfactory, h is halved and )] ( ), ( [ t u t x f is evaluated 4 times again and tested for accuracy again. The procedure is repeatedly until the accuracy specified by the user is met. If the accuracy is too high, h is doubled. If the satisfactory step-size is 1 h , the next step begins with time 1 0 h t t + = and the numerical integration predicts ) ( 2 1 0 h h t x + + . This continues until the end-time of the integration specified by the user, at which point ) .... ( 2 1 0 N h h h t x + + + is predicted. A priori, it is not possible to know what step-sizes to use. Therefore, some exploratory tests are required. This consists of carrying on the simulation with an educated guess of h . Then h is reduced and if the second simulation shows that there are discrepancies, h must be reduced again and again until one is satisfied that h is fine enough. Of course, one can carry on the reverse test by increasing h if there is reason to believe that it is too fine and therefore wastefully time consuming. Apart from accuracy, using too large a step size can meet numerical instability. Most of

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