First thing, you could have mentioned, what RK method you have used. Here is a brief introduction to RK methods and Euler method, working, there merits and demerits.
Euler method
Euler method is first order method. It is straight forward method that evaluate the next point based on slope information of current point and easy to code. It is a single step method. Stability of Euler method is relatively small. For complex problem and (or) boundary condition it may fail. It can be used for basic numerical analysis. This method is not commonly used for spatial discretization but some times used in time discretization. This scheme is not recommended for hyperbolic differential equation because this is more diffusive. Order of convergence of this scheme with grid refinement is very poor. Extending Euler method to higher order method is easy and straight forward.
RK methods:
If you consider Runge Kutta that is not a single scheme, this is a family of schemes derived in specific style. You can refer this link to get a basic idea of RK method http://web.mit.edu/10.001/Web/Course_Notes/Differential_Equations_Notes/node5.html
RK method is multi-stage method because it involves slope calculations at intermediate distance between current grid to next grid. It calculates next the value of dependent variable in next point by giving certain weight to slope at different location in between the current values point to next evaluating point based on Taylor series. RK method involves solving non-linear algebraic equations and that equations are formed by matching with Taylor series. Developing higher order RK methods is tedious and almost impossible without using symbolic tools in computation. Most famous RK method is RK4 because of its good convergence rate and stability property.
Answer
Usually error in Euler method is higher than three stage or more RK
method (RK3), because truncation error in RK methods is less compared to Euler method.In most of the beginner level litterateurs in numerical methods, it is wrongly mentioned that higher order methods (say, RK4) give less error than lower order method (say, Euler method). Most of the time this may be true (not all the time). This property is completely depends on mesh and differential equations you have considered.
Initial error in RK4 method is higher than Euler method for coarse grid and reduces with refining grid, because convergence rate of RK4 method is more than Euler. As a thump rule we can say Euler methods works better than RK4 method in coarse grid and RK4 method is good in fine grid. Please note that coarseness or fineness of grid is completely based on differential equation, initial condition and numerical scheme.
You can try this experiment in your code with different differential equations, different number of grids with Euler and RK4 if you have enough time.
