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| Highest-order derivative appearing in the ODE |
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A higher-order ODE can always be transformed into equivalent first-order system. For explicit kth order, define k new unknowns u1(t) = y(t), ..., uk(t) = y(k-1)(t).
Can write as system u' = g(t,u) where u1'=u2, ..., uk' = f(t, u1, ..., uk) |
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If f does not depend explicitly on t
Can be written in form y' = f(y) |
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| Linear Homogeneous Systems with Constant Coefficients |
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Linear - f has form f(t,y) = A(t)y + b(t) where A(t) and b(t) are matrix-valued and vector-valued functions of t
Homogeneous - b(t) = 0
Constant Coefficients - A does not depend on t
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| Solution to u' = f(y) is stable if for every ε > 0, there is a δ > 0 such that if y(t) satisifies the ODE and ||y(t) - u(t)|| <= δ then ||y(t) - u(t)|| <= ε for all t. |
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| Jacobian matrix for nonlinear system |
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| {Jf(t,y)}ij = ∂fi(t,y)/∂yj |
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| yk+1=yk+hk(f(tk,yk) + f(tk+1, yk+1))/2 |
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| Due to the method used, and would remain even if all arithmetic were performed exactly |
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| The error made in one step of the numberical method |
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| Cumulative overall error. The error at step k is the difference between the approximate solution, yk, and the true solution y(xk). |
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| Accuracy of a numerical method is of order p if lk=O(hkp+1) |
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| Stability of Numerical Method |
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| Small perturbations do not cause the resulting numerical solution to diverge away without bound |
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| Euler's Method for Scalar ODE y'=λy |
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yk+1=yk+h λyk
yk+1=(1+ hλ)ky0
So (1+ hλ)k is called the growth factor, and the magnitude of 1+ hλ must be less than one with Re(λ)<0, otherwise the method is unstable |
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| The global error is not simply the sum of the local errors. If the solutions of the ODE are diverging, the local errors at each step are magnified over time, so that global error is greater than sum of local error. If solutions of ODE are converging then global error may be less than that sum fo the local errors. |
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y' = λy, y(0) = y0
has exact solution y(t) = y0eλt
Euler's Method: yk+1 = yk + λhyk = (1 + λh)y0
So, (1 + λh) is an amplification factor (must be less than 1 in magnitude) |
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| Use information at more than one previous point to estimate the solution at the next point. Need to use some other method to obtain the initial point. |
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| Adams Explicit Method (Adams-Bashforth predictor) |
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| Adams Implicit Method (Adams-Moulton Corrector) |
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| A good initial guess is conveniently supplied by an explicit method so the explicit and implicit methods are used together. A fixed number of corrector steps (often only one) can be used to reduce the expense of implicit methods. |
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Derived by interpolating derivative values y' = f at m previous points and then integrating the resulting interpolating polynomial to obtain
[image] |
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| Backwards Differentiation Formula Methods |
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| Derived by interpolating the solution values y at m previous points, differentiating the resulting interpolating polynomial, and setting the derivative equal to f(tk+1, yk+1) at tk+1 to obtain yk+1. |
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Retaining more terms in the Taylor series, we can generate higher-order single-step methods than Euler's method
e.g.: yk+1=yk+hkyk' + hk2/2 yk'' |
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| Single-step methods, similar to Taylor series methods, but replace higher derivatives by finite difference approximations based on values of f at points between tk and tk+1 |
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| Based on use of a single-step method to integrate the ODE over a given interval using several different step sizes and yilding results denoted by Y(hi). Gives discrete approximate to Y(h) where Y(0) = y(tk+1). Fit interpolating polynomial to these data, and approximate Y(0). |
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