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According to the Gf Simmons book, the local truncation error in fourth order Runge-Kutta is $(1/180)y^{(5)}(a)h^5$, but according to me it is $(1/3840)y^{(5)}(a)h^5$

I used Taylor expansion to do that,i am attaching the picture of that part of book as well as my method.Thanks in advance for helping me with enter image description herethe doubt.

Lutz Lehmann
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Sachin_1729
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  • This is both wrong, the leading error term in RK4 is not a pure derivative., not a multiple of the 5th derivative. – Lutz Lehmann Apr 01 '23 at 20:16
  • In multi-step methods like Adams-Bashford and Adams-Moulton you get full derivatives as leading error coefficients, and they may look like the ones you cite. But then that is something that your source got wrong. – Lutz Lehmann Apr 01 '23 at 20:56

1 Answers1

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It is correct that the expansion of the error term starts with $e(t,h)=c_5(t)h^5+...$, but both expressions for the coefficient are wrong, the leading coefficient in RK4's error is not a pure derivative, nor a multiple of the 5th derivative.

Expressed in combinations of the ODE function, the exact 5th derivative of the ODE solution expressed is $$ \frac{y^{(5)}(x)}{5!} = \frac{f_{4}[f_{0}^4]}{120} + \frac{f_{3}[f_{1}[f_{0}],f_{0}^2]}{20 } + \frac{f_{2}[f_{2}[f_{0}^2],f_0]}{30} + \frac{f_{2}[(f_{1}[f_{0}])^2]}{40} + \frac{f_{2}[f_{1}[f_{1}[f_{0}]],f_{0}]}{30} + \frac{f_{1}[f_{1}[f_{1}[f_{1}[f_{0}]]]]}{120} + \frac{f_{1}[f_{1}[f_{2}[f_{0}^2]]]}{120} + \frac{f_{1}[f_{3}[f_{0}^3]]}{120} + \frac{f_{1}[f_{2}[f_{1}[f_{0}],f_{0}]]}{40} $$ where $f_0=f(x)$, $f_k=f^{(k)}(x)$, the derivative tensors, are vector-valued symmetric multilinear forms, the arguments in $[,]$ are the vector entries to these multilinear forms, powers here indicate repetition of the argument.

The first error term of RK4 in contrast is $$ \frac{y_1-y(x_1)}{h^5} = \frac{f_{4}[f_{0}^4]}{2880} + \frac{f_{3}[f_{1}[f_{0}],f_{0}^2]}{480} - \frac{f_{2}[f_{2}[f_{0}^2],f_0]}{480} + \frac{f_{2}[(f_{1}[f_{0}])^2]}{160} + \frac{f_{2}[f_{1}[f_{1}[f_{0}]],f_{0}]}{120} - \frac{f_{1}[f_{1}[f_{1}[f_{1}[f_{0}]]]]}{120} + \frac{f_{1}[f_{1}[f_{2}[f_{0}^2]]]}{480} - \frac{f_{1}[f_{3}[f_{0}^3]]}{720} - \frac{f_{1}[f_{2}[f_{1}[f_{0}],f_{0}]]}{240} $$ Alone that there are mixed signs shows that this can not be a multiple of $y^{(5)}$.

For the order conditions expressed in full partial derivatives see Prove that Runge Kutta Method (RK4) is of Order 4. The terms were computed using manual post-processing of the results of Analytical expressions for ODE integration methods using Sympy

Lutz Lehmann
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  • Thank you,sir

    Please clarify this: do you mean implicit and explicit by multistep and non multistep methods,also according to what i understood Euler's method is multistep while it's improvement i;e Heun's method isn't

     – Sachin_1729 Apr 01 '23 at 22:09
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    Yes, the Euler methods are both, one-step and linear multi-step. Among the second order methods only the implicit trapezoidal method belongs to both classes. // And the 4th order AB and AM methods have different leading error terms, $\frac5{720}$ and $-\frac1{720}$ times $h^5y^{(5)}$. – Lutz Lehmann Apr 01 '23 at 22:29
  • Thank you sir,for your help – Sachin_1729 Apr 02 '23 at 04:41