Integration of 3-body encounter. Figure taken from

4. Integration of Ordinary Differential Equations (ODE). We consider ODE with one variable, Most of results for this single ODE can be applicable for the above n- coupled ODE.

Existence, uniqueness, and stability of solution for ODE, A quick look for

Euler’s method: Consider the initial value problem,

Error analysis for Euler’s method

Stability of Euler’s method

Taylor’s method:

Exc 4-1) has a solution x(t) = t ( 1+ ln t). Apply Euler’s method changing the h = 1/2 n, n=1 to 8, and estimate (a) the absolute error |y i - w i | at t = 6, and (b) the error ratio for the successive h = 1/2 n at the same instant t = 6, namely, Exc 4-2)Perform the same analysis as 4-1) using the second order and fourth order Taylor methods. Because computations of higher derivatives is cumbersome, higher-order formula which involves evaluation f(t,y) only is more convenient.

Summary of the key concept on numerical method for ODE. Global discretization error : Local truncation error  i : The amount that the solution of ODE fails to satisfy the the finite difference equation. ex.) One-step method. Definitions: (Consistency, Convergence and Stability)

One-step method. Theorem: (Convergence and stability of the one-step method.) One-step method is consistent if Remark: Above theorem says the one-step method is consistent ) convergent. It can be proved under the same conditions, the one-step method is convergent, consistent.

One-step method. 2 nd order Runge-Kutta method. Determine constants a 1, a 2,  2,  2, so that  becomes O(h 2 ) approximation of the O(h 2 ) Taylor method. Heun method. (One step Euler + Trapezoidal integration.) Modified Euler method. (Half step Euler + Midpoint integration.)

Classical 4 th order Runge-Kutta method. Exc 4-3) Using an algebraic computing software, show that the local truncation error of Classical 4 th -order Runge-Kutta method is O(h 4 ). Optimal RK2 method.

General s-stage Runge-Kutta method.

Remarks on General s-stage Runge-Kutta method. (1) For Explicit s-stage formula, it is not known in general what order O(h p ) of formula one can construct for each level s. Formulas with properties (1) Small local truncation error, (2) Coefficients to be rational numbers, (3) many zeros in a j,l, are more practical. Exc 4-4) Using the idea of Gauss-Legendre integration formula, derive 2- stage Runge-Kutta formula with 4 th order accuracy. (2) For Implicit s-stage formula, it is known that O(h 2s ) formula can be construct for each level s.

Error control: Estimate the local truncation error and make it smaller than a certain threshold value by changing step size. How to estimate the error ? 1) Just to make h ! h/2. This is fine, but inefficient. 2) Embedded formula. : for RK 8 th order formula.

Linear (m-step) Multistep method. Substitute the form f(t,y(t)) = p(t) + R(t) in the integral form of ODE, and integrate it to calculate b j. Implicit linear multistep method is derived from interpolating polynomial of order m. Adams method: a j = 0, for j = 2,.., m Derivation: Integrating the both side of ODE, Explicit linear multistep method from interpolating polynomial of order m-1

Explicit scheme is also called Adams-Bashforth, implicit Adams-Moulton. Explicit scheme may be efficient since the f(t i,w i ) of earlier steps are used. The starting values for w 0 (initial value), w 1, …, w m-1 are required for the m-step method. These are calculated from one-step method of the same order. For the implicit method, value at t i+1, w i+1, is calculated from an algebraic equation. It is iteratively solved using w i+1 of an explicit multistep method of the same order as an initial guess. Usually this iteration is done by direct substitution, and only one or two iteration is made. This procedure is called predictor – corrector schemes. For this scheme, the 4 th -order formula is the most popular.

Exc4-5) Assuming uniform discretization in t domain, derive linear 2-step, 3- step 4-step explicit formulas and 1-step, 2-step, 3-step implicit formulas. Compare the coefficients of error terms between implicit and explicit formulas of the same order. In the Adams method, Newton form of polynomial interpolation formula is used for changing the step size as well as the starting formula. (Krogh type formula.) The local truncation error estimation is often made by the difference between predictor value and corrector value, which can be used for control the step size h. Exc4-6) Report on Krogh type formula. Exc4-7) Report on the ODE solver that uses Richardson extrapolation.

Conditions for the coefficients a j, and b J to satisfy for having the local truncation error  i = O(h p ) are written Order of General Linear (m-step) Multistep method. Exc4-8) Derive these conditions. hint) Substitute dy/dt = f(t,y) to the equation for  i above, and expand y and y’ around t* = t i+1 - m.

Consistency: (  i ! 0, as h ! 0) is satisfied if the local truncation error  i is a at least O(h). Consistency Convergence, and Stability of General Linear Multistep method. Note: the starting values w 1, …, w m-1 are assumed to be converge.

Stability: Consider the case with f(t,y) = 0. Definition: (Root condition) The linear multistep method satisfies the Root condition if the zeros of the associated characteristic polynomial satisfy Theorem: (Stability) A linear multistep method is stable if and only if it satisfies the root condition. Definition: A stable multistep method is said to be strongly stable if = 1 is the only zero of P( ) with | | = 1, and to be weakly stable otherwise. Remark: A linear multistep method that satisfies the consistency condition will always have at least one zero with =1.

Convergence: Theorem: (Convergence) A linear multistep method is convergent, if and only if it is both consistent and stable

Absolute stability and stiff equations. Test problem: This problem models a system of linear ODEs, in which case represents an eigenvalue of the Jacobian associated with the r.h.s. When the asymptotic character of a numerical approximation w n, n ! 1 matches that of exact solution y(t), t ! 1, the numerical method is said to be absolute stable. One-step method: Consider the m th order Taylor methods. Definition: The region of absolute stability for a one-step method is the set

Multistep method: Consider the linear m-step multistep method. Definition: The region of absolute stability for a multistep method is the set Exc 4-9) Derive the roots  k for 2-step Adams-Bashforth method and 2-step Adams Moulton method, and show the region of absolute stability on the complex plane.

Stiffness ratio: Suppose { k } is the set of characteristic expoennts associated with a particular ODE, stiffness ratio is defined by Methods to efficiently compute a solution of stiff ODE are required to have regions of absolute stability as large as possible. Possibly whole z < 0 plane. Definition: (A-stable, Dahlquist) A numerical method is A-stable if it is absolutely stable for all h such that Re h  Some known results: Explicit Runge-Kutta methods are not A-stable. Explicit linear multistep methods are not A-stable. Order of A-stable implicit multistep methods is less than 2. Among the A-stable implicit multistep methods, the trapezoidal method has the smallest local truncation error. Any symmetric s-stage and O( h 2s ) order implicit Runge-Kutta method (implicit Gauss formula) are A-stable. Also Radau, Lobatto formulas.