Finite-budget positive-tube forward Euler loop

Fix the periodic SBP data and semidiscrete vector field $L_h$ of Periodic diagonal-norm SBP compressible viscous scheme. Let $\mathcal{C}\subset \mathcal{G}$ be a specified compact set, let the initial nodal vector $U^0=(U_i^0{)}_{i\in \mathcal{I}}$ satisfy $U_i^0\in \mathcal{C}$ for every node, let $\mathrm{\Delta }t>0$, and let $N\in \mathbb{N}\cup \{0\}$ be a finite step budget. The algorithm returns a nodal vector, a status, and an accepted-step count.

1. For $n=0,\dots ,N-1$, compute the candidate $U^{n+1,\ast }=U^n+\mathrm{\Delta }t{L}_h(U^n)\mathrm{.}$

2. If $U_i^{n+1,\ast }\in \mathcal{C}$ for every node $i$, accept it by setting $U^{n+1}=U^{n+1,\ast }$ and continue.

3. If any candidate node lies outside $\mathcal{C}$, return $(U^n,\mathtt{\text{left\_positive\_tube}},n)$ immediately. Here $U^n$ is the last accepted vector and $n$ is the number of accepted steps; the rejected candidate is the one-based stage $n+1$.

4. If all $N$ candidates are accepted, return $(U^N,\mathtt{\text{completed}},N)$. For $N=0$, return $(U^0,\mathtt{\text{completed}},0)$.

The membership test is applied nodewise to the compact positive-state tube. A returned left_positive_tube status records a failed candidate and makes no claim that the prescribed step budget was completed.