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• Usually error in Euler method is higher than higher order RK method (RK2, RK3, etc.), because truncation error in higher order methods is less compared to Euler method.

• In some of the beginner level literature in numerical methods, it is loosely mentioned that higher order methods (say, RK4) give less error than lower order method (say, Euler method). Most of the time this is true, but not all the time. This property depends on the mesh and initial condition and differential equations you have considered.

• If the exact solution to the differential equation is a polynomial of order $n$, it will be solved exactly by an $n$-th Runge-Kutta method. For example, forward Euler will be exact if the solution is a line. RK4 will be exact if the solution is a polynomial of degree 4 or less.

• Initial "absolute maximum difference error" in RK4 method is equal (or) higher than Euler method for coarse grid and reduces with refining grid for problems with shorter waves relative to grid. Because convergence rate of RK4 method is more than Euler. Please note that coarseness or fineness of grid is completely based on differential equation, initial condition and numerical scheme. Please refer following link for more details. Though that is based on differentiation, we can make a relative comparison between time numerical integration and differentiation as long as the "numerical integration" is stable.

• Usually error in Euler method is higher than higher order RK method (RK2, RK3, etc.), because truncation error in higher order methods is less compared to Euler method.

• In some of the beginner level literature in numerical methods, it is loosely mentioned that higher order methods (say, RK4) give less error than lower order method (say, Euler method). Most of the time this is true, but not all the time. This property depends on the mesh and initial condition and differential equations you have considered.

• If the exact solution to the differential equation is a polynomial of order $n$, it will be solved exactly by an $n$-th Runge-Kutta method. For example, forward Euler will be exact if the solution is a line. RK4 will be exact if the solution is a polynomial of degree 4 or less.

• Initial "absolute maximum difference error" in RK4 method is equal (or) higher than Euler method for coarse grid and reduces with refining grid for problems with shorter waves relative to grid. Because convergence rate of RK4 method is more than Euler. Please note that coarseness or fineness of grid is completely based on differential equation, initial condition and numerical scheme. Please refer following link for more details. Though that is based on differentiation, we can make a relative comparison between time numerical integration and differentiation as long as the "numerical integration" is stable.

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. Order of convergence of scheme can be find out by calculating slope of plot - $log(error)$ vs $log(grid size)$.

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. Order of convergence of scheme can be find out by calculating slope of plot - $\log(\mathrm{error})$ vs $\log(\mathrm{grid\ size})$.

• Usually error in Euler method is higher than higher order RK method (RK2, RK3, etc.), because truncation error in higher order methods is less compared to Euler method.

• In some of the beginner level literature in numerical methods, it is loosely mentioned that higher order methods (say, RK4) give less error than lower order method (say, Euler method). Most of the time this is true, but not all the time. This property depends on the mesh and initial condition and differential equations you have considered.

• If the exact solution to the differential equation is a polynomial of order $n$, it will be solved exactly by an $n$-th Runge-Kutta method. For example, forward Euler will be exact if the solution is a line. RK4 will be exact if the solution is a polynomial of degree 4 or less.

• Initial "absolute maximum difference error" in RK4 method is equal (or) higher than Euler method for coarse grid and reduces with refining grid, because convergence rate of RK4 method is more than Euler. Please note that coarseness or fineness of grid is completely based on differential equation, initial condition and numerical scheme. Please refer following link for more details. Though that is based on differentiation, we can make a relative comparison between time numerical integration and differentiation as long as the "numerical integration" is stable.

• Usually error in Euler method is higher than higher order RK method (RK2, RK3, etc.), because truncation error in higher order methods is less compared to Euler method.

• In some of the beginner level literature in numerical methods, it is loosely mentioned that higher order methods (say, RK4) give less error than lower order method (say, Euler method). Most of the time this is true, but not all the time. This property depends on the mesh and initial condition and differential equations you have considered.

• If the exact solution to the differential equation is a polynomial of order $n$, it will be solved exactly by an $n$-th Runge-Kutta method. For example, forward Euler will be exact if the solution is a line. RK4 will be exact if the solution is a polynomial of degree 4 or less.

• Initial "absolute maximum difference error" in RK4 method is equal (or) higher than Euler method for coarse grid and reduces with refining grid for problems with shorter waves relative to grid. Because convergence rate of RK4 method is more than Euler. Please note that coarseness or fineness of grid is completely based on differential equation, initial condition and numerical scheme. Please refer following link for more details. Though that is based on differentiation, we can make a relative comparison between time numerical integration and differentiation as long as the "numerical integration" is stable.