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[This is the picture Lutz Sir] 1For single differential equation With y'=f(x,y) and y(x_0)=y_0 ,it is x_(k+1)=x_(k)+(1/6)(m1+2m2+2m3+m4)

For higher order of differential equations it is,when converted to system of differential equations

x'=f(t,x,y) and y'=g(t,x,y)=x(in case of higher de being converted to system of differential equations) and x(t_0)=x_0 and y(t_0)=y_0

X_(k+1)=x_(k)+(1/6)(mk1+mk2+mk3+mk4) And for y_(k+1)=y_(k)+(1/6)(nk1+nk2+nk3+nk4)

Why so?

Where did the 2 of the 2nd and 3rd term disappear and why?

All the mk's and nk's and M's denote the usual terms in runge kutta method???

Sachin_1729
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  • Do you have a source for that? The final composition is also in the vector/system case based on the Simpson method, the coefficients must at least sum to one. Meaning the factors 2 should be there. – Lutz Lehmann Apr 02 '23 at 18:55
  • Yes sir,the source is George simmons differential book(last chapter) i am updating the question and attaching the picture also – Sachin_1729 Apr 02 '23 at 19:31
  • There are two Simmons books, the earlier with a theoretical and historical focus, the second with Krantz a compilation of extracts of the first with numerical methods added in. It seems that at places details and proper context got lost. – Lutz Lehmann Apr 02 '23 at 20:13
  • It is from the first one(without krantz as the author),what is the answer to above question???is it given wrong – Sachin_1729 Apr 03 '23 at 00:49
  • Yes, it is wrong. There is no difference between the scalar and system variants, one could see the scalar variant as system variant for dimension one. See my answer in https://math.stackexchange.com/questions/528856/explanation-and-proof-of-the-4th-order-runge-kutta-method/4666744#4666744 on how the stages of RK4 can be interpreted. – Lutz Lehmann Apr 03 '23 at 05:17
  • Thank you sir,i was thinking about it from few days.I read your answer and i am convinced now. – Sachin_1729 Apr 03 '23 at 07:48

1 Answers1

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All the well-known methods are designed to work for scalars as well as for vector-valued ODE or ODE systems. That is, for the application it makes no difference if the $x$ in $\dot x=f(x)$ is a scalar or a vector, and $f$ correspondingly a scalar function or a system of differential equations.

This property is not universally true. There are Runge-Kutta methods where one gets different error orders, the scalar variant having a higher order than the vector variant. This is due to the fact that in the scalar case all derivatives are scalar valued and thus commute. In the system case the derivatives are tensors, vector-valued symmetric multi-linear forms where commutativity does not make much sense. For example, $f'(x)f''(x)[f(x),f(x)]$ is different from $f''(x)[f'(x)f(x),f(x)]$. Expressed in partial derivatives with summation convention $$ \frac{\partial f_i}{\partial x_j} \frac{\partial^2f_j}{\partial x_k\partial x_l} f_kf_l ~~\text{ vs. }~~ \frac{\partial^2f_i}{\partial x_j\partial x_k} \frac{\partial f_j}{\partial x_l} f_lf_k $$

Or one order higher, $$ f'f''[f'f,f],~~ f''[f'f'f,f]~~\text{and}~~ f''[f'f,f'f] $$ are all different in the non-scalar case. This means that certain order conditions for the method coefficients that are different for the vector case combine into one if derived strictly for the scalar case.

The most famous example is the 5th order 5-stage method that Martin Kutta gave in his original paper in 1901. Such a method with these properties is impossible in the system case.

In summary:

  • Any change in the coefficients gives a different method, any contrary claim is wrong, it is an error either in the naming of the method or in the coefficients presented.

  • Any method has to be at least zero-order consistent. Meaning that the integration of $\dot x=f(x)=1$ has to result in $x(t)=t+x_0-t_0$ without any errors. This implies that the sum of the combination coefficients for the result of the step has to be $1$, and $\frac16(1+1+1+1)=\frac23$ as per the presented source is not one.

Lutz Lehmann
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