Parameters
ODE Type
Decay rate k
1.00
Initial value y₀
1.00
Step size h
0.10
Range: 0.001 to 1.0 (log scale)
Time span T
10.0
Order of Accuracy
Global Error:
Euler: $\mathcal{O}(h)$ | RK2: $\mathcal{O}(h^2)$ | RK4: $\mathcal{O}(h^4)$
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Euler Max Error
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RK2 Max Error
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RK4 Max Error
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Steps
Global Error |y_num − y_exact| vs t
Method Formulas
Euler (1st order):
$$y_{n+1} = y_n + h\,f(t_n,\,y_n)$$Heun / RK2 (2nd order):
$$k_1 = f(t_n,y_n),\quad k_2 = f(t_n+h,\,y_n+hk_1)$$ $$y_{n+1} = y_n + \tfrac{h}{2}(k_1+k_2)$$Runge-Kutta 4th order:
$$k_1=f(t_n,y_n),\; k_2=f\!\left(t_n+\tfrac{h}{2},y_n+\tfrac{h}{2}k_1\right)$$ $$k_3=f\!\left(t_n+\tfrac{h}{2},y_n+\tfrac{h}{2}k_2\right),\; k_4=f(t_n+h,y_n+hk_3)$$ $$y_{n+1}=y_n+\tfrac{h}{6}(k_1+2k_2+2k_3+k_4)$$
CAE Relevance: FEM time-integration schemes (Newmark-β, central difference) are direct applications of ODE solvers. LS-DYNA's explicit solver requires Δt < Δtcr = 2/ωmax for stability. RK4 is used as a reference algorithm in implicit solvers such as ABAQUS.