2D Trapezoidal and Simpson Rules
Source inspiration: (Mathew 2000-2019).
Description
This page studies numerical approximation of a rectangular double integral with the composite 2D Trapezoidal and 2D Simpson rules. The archived module examples use
\[ I = \int_0^1 \int_0^1 8e^{-x^2-y^4}\,dy\,dx, \]
with surface \(z=8e^{-x^2-y^4}\) over \(R=\{(x,y):0\le x\le 1,\ 0\le y\le 1\}\).
For a tensor-product grid, the 2D Trapezoidal rule applies endpoint/interior weights along each axis and sums weighted samples:
\[ I_{T} \approx hk\sum_{i=0}^{m}\sum_{j=0}^{n} w_i w_j f(x_i,y_j), \qquad w_0=w_m=\tfrac{1}{2},\ w_i=1\ (1\le i\le m-1), \]
and the 2D Simpson rule uses the product Simpson weights:
\[ I_{S} \approx \frac{hk}{9}\sum_{i=0}^{m}\sum_{j=0}^{n} a_i a_j f(x_i,y_j), \qquad a_0=a_m=1,\ a_i=4\ \text{(odd i)},\ a_i=2\ \text{(even i)}. \]
The archived snapshot is incomplete for some intermediate image IDs, so these animations are reconstructed from the available formula and checkpoint assets.
Animations
Each animation shows full-domain mesh refinement on \([0,1]\times[0,1]\) for \(f(x,y)=8e^{-x^2-y^4}\), using the archived benchmark \(I\approx 5.04756680717\).
Julia source scripts that generated these animations are linked under each case.
Case 1 - 2D Trapezoidal Rule, \(f(x,y)=8e^{-x^2-y^4}\)
Behavior: The coarse-to-fine tensor grid updates over the whole square each frame. The approximation converges toward the archived reference value, with visible second-order error decay under refinement.
Case 2 - 2D Simpson Rule, \(f(x,y)=8e^{-x^2-y^4}\)
Behavior: Simpson tensor-product weights are recomputed on each refined mesh. Compared with trapezoidal refinement, the Simpson estimate reaches high accuracy with fewer refinements.
