Lecture 2 - 2 Multistep Methods Up to now all methods we...

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2 Multistep Methods Up to now, all methods we studied were single step methods, i.e., the value y n +1 was found using information only from the previous time level t n . Now we will consider so-called multistep methods , i.e., more of the history of the solution will affect the value y n +1 . 2.1 Adams Methods Consider the first-order ODE y ( t ) = f ( t, y ( t )) . If we integrate from t n +1 to t n +2 we have t n +2 t n +1 y ( τ ) = t n +2 t n +1 f ( τ, y ( τ )) or y ( t n +2 ) - y ( t n +1 ) = t n +2 t n +1 f ( τ, y ( τ )) dτ. (31) As we saw earlier for Euler’s method and for the trapezoidal rule, different numer- ical integration rules lead to different ODE solvers. In particular, the left-endpoint rule yields Euler’s method, while the trapezoidal rule for integration gives rise to the trapezoidal rule for IVPs. Incidentally, the right-endpoint rule provides us with the backward Euler method. We now use a different quadrature formula for the integral in (31). Example Instead of viewing the slope f as a constant on the interval [ t n , t n +1 ] we now represent f by its linear interpolating polynomial at the points τ = t n and τ = t n +1 given in Lagrange form, i.e., p ( τ ) = τ - t n +1 t n - t n +1 f ( t n , y ( t n )) + τ - t n t n +1 - t n f ( t n +1 , y ( t n +1 )) = t n +1 - τ h f ( t n , y ( t n )) + τ - t n h f ( t n +1 , y ( t n +1 )) , where we have used the stepsize t n +1 - t n = h . FIGURE Therefore, the integral becomes t n +2 t n +1 f ( τ, y ( τ )) t n +2 t n +1 p ( τ ) = t n +2 t n +1 t n +1 - τ h f ( t n , y ( t n )) + τ - t n h f ( t n +1 , y ( t n +1 )) = f ( t n , y ( t n )) - 1 2 ( t n +1 - τ ) 2 h + f ( t n +1 , y ( t n +1 )) ( τ - t n ) 2 2 h t n +2 t n +1 = 3 h 2 f ( t n +1 , y ( t n +1 )) - h 2 f ( t n , y ( t n )) . 42
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Thus (31) motivates the numerical method y n +2 = y n +1 + h 2 [3 f ( t n +1 , y n +1 ) - f ( t n , y n )] . (32) Since formula (32) involves two previously computed solution values, this method is known as a two-step method . More precisely, is is known as the second-order Adams- Bashforth method (or AB method) dating back to 1883. Remark 1. We will establish later that this method is indeed of second order ac- curacy. 2. Note that the method (32) requires two initial conditions. Since the IVP will give us only one initial condition, in the Matlab demo script ABDemo.m we take the second starting value from the exact solution. This is, of course, not realistic, and in practice one often precedes the Adams-Bashforth method by one step of, e.g., a second-order Runge-Kutta method (see later). However, even a single Euler step (which is also of order O ( h 2 )) can also be used to start up (and maintain the accuracy of) the second-order AB method. This approach can also be used in ABDemo.m by uncommenting the corresponding line. Example The Matlab script ABDemo.m compares the convergence of Euler’s method (the one-step AB method) with the two-step AB method (32) for the IVP y ( t ) = - y 2 ( t ) , y (0) = 1 on the interval [0 , 10] with different stepsizes N = 50 , 100 , 200 and 400. The exact solution of this problem is y ( t ) = 1 t + 1 .
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