Periodic semi-Lagrangian error is controlled by local foot and interpolation defects

Fix $T>0$ and a periodic box with grid width $h={\max }_j{h}_j$. Let $u\in C^1([0,T];C^2({\mathbb{T}}_{\mathrm{\ell }}^d))$ solve the nonconservative linear transport equation $u_t+b\cdot \mathrm{\nabla }u=0,$ with the periodic $C^1$ velocity and global flow of A periodic C1 velocity has a global characteristic flow on a finite horizon. Consider $N$ equal time steps $\mathrm{\Delta }t=T\mathrm{/}N$ and node updates $U_{\mathit{i}}^{n+1}=({\mathcal{J}}_h{U}^n)({\tilde{x}}_{\mathit{i}}^n),$ where ${\mathcal{J}}_h$ reconstructs a nodal array as a periodic function and ${\tilde{x}}_{\mathit{i}}^n$ is a computed backward foot. Suppose constants $\mathrm{\Lambda }\ge 1$, $C_I\ge 0$, and $C_F\ge 0$ are uniform over the refinement family and independent of $h,\mathrm{\Delta }t,N$. Assume $\mathrm{\parallel }{\mathcal{J}}_{h}V-{\mathcal{J}}_{h}W{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le \mathrm{\Lambda }\mathrm{\parallel }V-W{\mathrm{\parallel }}_{{\mathrm{\ell }}^{\mathrm{\infty }}}$ for all node arrays, the pointwise interpolation defect at each computed foot is at most $C_I{h}^{r+1}$ for the exact samples $u_{\mathit{i}}^n=u(x_{\mathit{i}},t_n)$, and, for compatible lifts of the computed and exact feet, the foot error satisfies $\mathrm{\parallel }{\tilde{x}}_{\mathit{i}}^n-X(t_n;t_{n+1},x_{\mathit{i}}){\mathrm{\parallel }}_2\le C_F(\mathrm{\Delta }t{)}^{q+1}$, for integers $q,r\ge 1$. If $L_u:={\sup }_{[0,T]\times {\mathbb{T}}_{\mathrm{\ell }}^d}\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2$ and $E_n:={\max }_{\mathit{i}}\mathrm{\mid }{U}_{\mathit{i}}^n-u(x_{\mathit{i}},t_n)\mathrm{\mid }$, then, with ${\delta }_h:=C_I{h}^{r+1}+\mathrm{\Lambda }{L}_u{C}_F(\mathrm{\Delta }t{)}^{q+1},$ $E_N\le {\mathrm{\Lambda }}^N{E}_0+{\delta }_h{\sum }_{k=0}^{N-1}{\mathrm{\Lambda }}^k\mathrm{.}$ In the contractive case $\mathrm{\Lambda }=1$, this becomes $E_N\le E_0+T(C_I\frac{h^{r+1}}{\mathrm{\Delta }t}+L_u{C}_F(\mathrm{\Delta }t{)}^q)\mathrm{.}$ For the explicit Euler and tensor multilinear algorithm, Explicit Euler gives a second-order local backward-foot defect, Tensor multilinear interpolation has a second-derivative error bound, and Periodic multilinear interpolation is convex and L-infinity contractive give $q=r=1$ and $\mathrm{\Lambda }=1$. Thus convergence for this branch follows, for example, when $E_0\to 0$, $\mathrm{\Delta }t\to 0$, and $h^2\mathrm{/}\mathrm{\Delta }t\to 0$. No classical advective CFL restriction is used; the joint consistency refinement is still required by this bound.

For $\mathrm{\Lambda }>1$, the geometric sum in the first estimate is not uniformly bounded as $N$ grows. A high-order interpolation or limiter must therefore supply its actual operator bound and local defect before this theorem yields a stable convergence claim. This theorem is for the periodic branch; a nonperiodic boundary branch must separately account for first-hit and boundary-data error as in A transverse backward boundary hit has a certified time and trace error bound.