Explicit Euler characteristic backtracking with multilinear interpolation

Inputs are a periodic grid from Periodic tensor grids and multilinear interpolation, a finite horizon $T>0$, a positive integer $M$ of time steps, a periodic $C^1$ velocity $b$, periodic initial data $u_0$, node values $U_{\mathit{i}}^0=u_0(x_{\mathit{i}})$, and a nonnegative integer step budget $B$. Write $\mathrm{\Delta }t=T\mathrm{/}M$ and $t_n=n\mathrm{\Delta }t$. The output is either success with $U^0,\dots ,U^M$, or budget-exhausted with no partial final state.

1. If $B<M$, return budget-exhausted before changing the input data.
2. For $n=0,\dots ,M-1$ and each arrival node $x_{\mathit{i}}$, compute the explicit backward Euler foot lift ${\tilde{x}}_{\mathit{i}}^n:=x_{\mathit{i}}-\mathrm{\Delta }tb(t_{n+1},x_{\mathit{i}})\mathrm{.}$ Reduce each coordinate modulo its period when evaluating the periodic interpolant.
3. Set $U_{\mathit{i}}^{n+1}:={\mathcal{I}}_h{U}^n({\tilde{x}}_{\mathit{i}}^n)\mathrm{.}$
4. After the $M$ finite updates, return success with all node arrays.

The finite step budget counts time updates. Every update evaluates one foot per grid node and a finite sum of $2^d$ interpolation terms. The procedure imposes no bound on $\mathrm{\parallel }b\mathrm{\parallel }\mathrm{\Delta }t\mathrm{/}h$; the floor-and-fraction convention in the interpolation definition applies even when a foot crosses many grid cells.

This algorithm is the periodic-box branch. It does not wrap a foot through a nonperiodic boundary. An inflow/exit implementation must instead use an explicit first-hit oracle and return an unresolved status when the first hit or its error certificate is not resolved; A transverse backward boundary hit has a certified time and trace error bound records the transverse error interface. No grazing, simultaneous, or unbounded event sequence is hidden in the periodic update.

The algorithm defines tensor-product multilinear interpolation only. A higher-order replacement needs its own operator-norm or limiter bound and its own interpolation defect estimate; high order alone does not imply $L^{\mathrm{\infty }}$ contractivity.