I want to solve a differential equation in the form
$$\frac{dx}{dt}=f(x,t)+g(x,t)$$
I divide this into two contributions as
$$\frac{dx_1}{dt}=f(x,t)$$
and
$$\frac{dx_2}{dt}=g(x,t)$$
I want to solve the first one by an adaptive step-size method (e.g. Runge-Kutta 4) and the second one by a fixed step-size method (e.g. Euler method). Now I have two questions:
1 - How should I handle different approaches of two methods with respect to the step size?
2 - How can I find the whole solution from the solutions of the parts? Is it just addition?
Your scheme doesn't succeed even in the simple case when $f(x,t)$ and $g(x,t)$ depend linearly over $x$. For instance $f(x,t)=x p(t), g(x,t)=x q(t) ~~~(2)$. then $$\frac{dx}{dt} =[p(t)+q(t)]x \implies \int \frac{dx}{x}=[p(t)+q(t)] dt \implies x=C_1 e^{\int [p(t)+q(t)] dt}~~~~(1)$$ If you solve $$\frac{dx_1}{dt}=x_1 p(t),~~ \frac{dx_2}{dt}= x_2 q(t) \implies x_1=C_2 e^{\int p(t) dt}.~~ x_2=C_3 e^{\int q(t) dt}.$$ Obviously, $$x=x_1+x_2=C_2 e^{\int p(t) dt}+C_3 e^{\int q(t) dt}~~~~(2) $$ (1) and (2) are diferent.