# Shared Flashcard Set

## Details

Initial Value Problems for ODEs
Eulers method; Accuracy and stability; Multistep methods; Taylor series, Runge-Kutta, extrapolation
27
Computer Science
03/04/2014

Term
 Order
Definition
 Highest-order derivative appearing in the ODE
Term
 Reduction to First Order
Definition
 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)
Term
 Autonomous Systems
Definition
 If f does not depend explicitly on t Can be written in form y' = f(y)
Term
 Linear Homogeneous Systems with Constant Coefficients
Definition
 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
Term
 Stability of Solutions
Definition
 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.
Term
 Jacobian matrix for nonlinear system
Definition
 {Jf(t,y)}ij = ∂fi(t,y)/∂yj
Term
 Euler's Method
Definition
 yk+1=yk+hkf(tk,hk)
Term
 Backward Euler's Method
Definition
 yk+1=yk+hkf(tk+1,yk+1)
Term
 Trapezoid Method
Definition
 yk+1=yk+hk(f(tk,yk) + f(tk+1, yk+1))/2
Term
 Truncation Error
Definition
 Due to the method used, and would remain even if all arithmetic were performed exactly
Term
 Local Error
Definition
 The error made in one step of the numberical method
Term
 Global Error
Definition
 Cumulative overall error.  The error at step k is the difference between the approximate solution, yk, and the true solution y(xk).
Term
 Order of Accuracy
Definition
 Accuracy of a numerical method is of order p if lk=O(hkp+1)
Term
 Stability of Numerical Method
Definition
 Small perturbations do not cause the resulting numerical solution to diverge away without bound
Term
 Euler's Method for Scalar ODE y'=λy
Definition
 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
Term
 Lipshitz Continuous
Definition
 [image]
Term
 Convergence of IVP ODE
Definition
 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.
Term
 Step Size Euler's Method
Definition
 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)
Term
 Multistep Methods
Definition
 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.
Term
Definition
 [image]
Term
Definition
 [image]
Term
 Predictor/Corrector Pair
Definition
 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.
Term
Definition
 Derived by interpolating derivative values y' = f at m previous points and then integrating the resulting interpolating polynomial to obtain [image]
Term
 Backwards Differentiation Formula Methods
Definition
 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.
Term
 Taylor Series Methods
Definition
 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''
Term
 Runge-Kutta Methods
Definition
 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
Term
 Extrapolation Methods
Definition
 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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