# Numerical Analysis/Order of RK methods/Quiz on RK order

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1

What is the main importance of having a higher ODE method order

 It always better to increase the accuracy of the solution It requires fewer steps for the same accuracy restriction We need the method to be better consistent We need the method to be more stable It provides better margin of stability

2

The number of slope predictions (number of k's) in an RK method is related to the order of the method as follows. The order of the method is:

 exactly equal to the number slope predictions equal to number of k's (slope evaluations) +1 upper bounded by the number of k's not related to the number of k's

3

An n-th order of the method means that

 the global truncation error is of the order ${\displaystyle O(h^{n+1})\,}$ the local truncation error is of the order ${\displaystyle O(h^{n})\,}$ the global truncation error is of the order ${\displaystyle O(h^{n})\,}$ the maximum of the global and local error is n-th order

4

The order of the local truncation error is related to the global truncation error in the following way

 they are about equal they are not related the global is one higher order than the local the local is one higher order than the global

5

Is the following true? The global truncation error is always less then the local, since the errors partially cancel out over steps.

 true false

6

What is the highest order that we can achieve with a Runge Kutta type method by using 5 k's?

 We can achieve 5-th order That method is not used, since it is not consistent. We can achieve 4-th order We can achieve even higher order, it is sixth order accurate.

7

How is the order of single step methods related to consistency of the method?

 At least first order of the method indicates that the method is consistent To be consistent, a method must be second order or higher not related at all, a single step method can be inconsistent and be of any order.

8

What is the main "tool" to prove the order of a numerical method for solving ODE's of type ${\displaystyle y'=f(t,y(t))\,}$?

 Solving a specific "stiff" problem ${\displaystyle y'=f(t,y(t))=-Ky\,}$ and showing that it converges for specific real positive K Taylor series expansion for one variable Taylor series expansion for one and two variables Solving the ODE equation analytically and then numerically

9

If we show that all terms in a numerical method for ODE recurrence equation match the terms in the Taylor series expansion up to the terms with ${\displaystyle h^{n}\,}$, but we do not analyze whether further cancellation of the terms exist, the following is true

 The method is of the order n The method is of the order n, since the global error is of the order ${\displaystyle O(h^{n})\,}$ The order of the method could be higher than n-1, but it is at least n-1

10

If we use an explicit n-th (n>1) order accurate multistep method with s steps (s>2) to solve an ODE, but we use an n-1 order method to calculate the missing initial points that we need to start using the multistep method, what is the global order of accuracy of our calculation?

 There is no such combined method, we cannot do that The overall method is n-th order, since we just use the n-1 order method for few initial points The overall method is of the order n-1, since the initial error from the lower accurate method remains in the cummulative error (global) The order of the method could be either n or n-1 The order of the method is something between n-1 and n, just like the RK 45 method

11

What do we get as an order of error from the following expression

${\displaystyle {\frac {1}{h}}O(h^{2})+10O(h^{3})\,}$?

 Something between ${\displaystyle O(h^{2})\,}$ and ${\displaystyle O(h^{3})\,}$ ${\displaystyle O(h^{2})\,}$ ${\displaystyle O(h^{3})\,}$ ${\displaystyle O(h)\,}$

12

What do we get as an order of error from the following expression

${\displaystyle \sin(h^{2}){\frac {1}{h}}O(h^{2})\,}$?

 Something between ${\displaystyle O(h^{2})\,}$ and ${\displaystyle O(h)\,}$ ${\displaystyle O(h^{2})\,}$ ${\displaystyle O(h^{3})\,}$ ${\displaystyle O(h)\,}$

13

What do we get as an order of error from the following expression

${\displaystyle e^{h}\cos(h){\frac {1}{h}}O(h^{2})\,}$?

 It must be of order ${\displaystyle O(h^{2})\,}$ ${\displaystyle O(h^{3})\,}$ ${\displaystyle O(h)\,}$