FA-87103 / Physics integrator stability / Member archive

Backward Euler drift uses the old velocity · case 03

Position lags the implicit velocity and energy is no longer monotone.

Member previewVariant 3 · 3 implementations · 8 checks per implementation

Case contract

solve(x0, v0, k, m, c, dt, steps): backward Euler for x'=v, m v'=-k x-c v in closed form v1=(v-dt*k*x/m)/(1+dt*c/m+dt^2*k/m), x1=x+dt*v1. Energy E=0.5 m v^2+0.5 k x^2; report the peak E_n/E_0 including step 0 (0 when E_0=0). Return [x, v, peak] rounded to 6.

Why this case matters

Game and robotics physics loops depend on integrator update order, step control and stabilization terms; a wrong decision point turns a stable simulation into drifting or exploding motion.

One recorded failure

Sample boundary fixture

This sample comes from the broken implementation of a controlled reproducer.

Boundary fixture	Actual	Expected	Outcome
case 0	`[-0.440518, 4.880917, 1.450295]`	`[-0.051052, 2.989657, 1.0]`	Failed

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