Robust series linearization of nonlinear advection-diffusion equations

Abstract

We consider nonlinear partial differential equations (PDEs) for advection-diffusion processes which are augmented by an auxiliary parameter $δ$ such that $δ=0$ corresponds to linear advection-diffusion. We derive potentially non-perturbative series expansions in $δ$ that provide a process to obtain the solution of the nonlinear PDE through solving a hierarchical system of linear, forced PDEs with the forcing terms dependent on solutions at lower orders in the hierarchy. We rigorously detail our approach for a particular deformation that interpolates between linear advection-diffusion and the canonical Burgers' equation modeling nonlinear advection. In this case, we prove that the series has infinite radius of convergence for arbitrary integrable initial data, analyze the cases of a Dirac-delta initial condition (IC) (i.e., the fundamental solution) in an infinite domain and arbitrary IC in a periodic domain, and demonstrate the approach to turbulent behavior in a scenario with periodic forcing. We then treat models of nonlinear diffusion involving the $p$-Laplacian operator, including generalizations of the Poisson equation in $1$ and $2$ dimensions, and the heat equation in $1+1$ dimensions. We detail series expansions for two different deformations of these equations about their linear (ordinary Laplacian) counterparts, providing numerical evidence for the convergence of the series outside of a perturbative regime and demonstrating that the rate and radius of convergence are affected by choice of deformation. Our results provide a rigorous foundation for using series expansion techniques to study nonlinear advection-diffusion PDEs, opening new pathways for analysis and potential applications for quantum-assisted computational fluid dynamics.

Figures (12)

• Figure 1: Convergence of series expansion for the fundamental solution of Burgers' equation. The exact solution of Burgers' corresponding to a Dirac-delta IC (i.e., \ref{['eq:BurgerLinearHomotopy_deltaIC']} with $\delta = 1$) and $R_e = 2$, is compared to the exact partial sum $\mathcal{S}_N (t,x;1)$ for varying expansion order $N$, time $t$, and (normalized) linear advection speed $v$. Left and right columns correspond to $v = 1$ and $v = 1/R_e$, respectively. Panels in the top row show the maximum spatial error between $u (t,x;1)$ and $\mathcal{S}_N (t,x;1)$ for $0 \leq N \leq 8$ and $0 < t \leq 10$. The solid black lines overlaying these plots indicate the error threshold of $\epsilon = 10^{-3}$, while the dashed black lines indicate a least-squares linear fit to a portion of this data. Middle and bottom rows show the exact solution compared to $\mathcal{S}_8$ for $t = 1$ and $t = 10$, respectively.
• Figure 2: Convergence of series expansion for the periodic Burgers' equation. The exact solution of Burgers' corresponding to a cosine-squared IC (i.e., Eq. \ref{['eq:BurgerLinearHomotopy_periodic_soln_cosinesquaredIC']} with $\delta = 1$) and $R_e = 500$, is compared to the exact partial sum $\mathcal{S}_N (t,x;1)$ for varying expansion order $N$, time $t$, and (normalized) linear advection speed $v$. Left and right columns correspond to $v = 1$ and $v = 1/R_e$, respectively. Panels in the top row show the maximum spatial error between $u (t,x;1)$ and $\mathcal{S}_N (t,x;1)$ for $0 \leq N \leq 8$ and $0 < t \leq 0.1$. The solid black lines overlaying these plots indicate the error threshold of $\epsilon = 10^{-3}$, while the dashed black lines indicate a least-squares linear fit to a portion of this data. Middle and bottom rows show the exact solution compared to $\mathcal{S}_8$ for $t = 0.01$ and $t = 0.1$, respectively.
• Figure 3: Convergence when refeeding is employed for periodic Burgers' equation. The exact solution of Burgers' corresponding to a cosine-squared IC (i.e., Eq. \ref{['eq:BurgerLinearHomotopy_periodic_soln_cosinesquaredIC']} with $\delta = 1$) and $R_e = 500$, is compared to the approximate series solution with and without refeeding for expansion order $N = 8$ and (normalized) linear advection speed $v = 1/R_e$. Left panel shows the maximum spatial error between $u (t,x;1)$ and $\mathcal{S}_N (t,x;1)$ versus $t$ with and without refeeding alongside a standard DNS approach to solving Burgers' equation. Middle panel shows the maximum spatial gradient of both the exact solution and $\mathcal{S}_8$ versus time. Right panel shows the exact solution compared to $\mathcal{S}_8$ with refeeding at the time where the $\max_x |\partial_x u (t,x;1)|$ is largest.
• Figure 4: Maximum spatial error versus expansion order for periodic Burgers' equation. The maximum spatial error between $u (t,x;1)$ and $\mathcal{S}_N (t,x;1)$ versus $N$ with refeeding employed at the time slices $t = 0.01$, $0.1$, $0.5$, and $1$.
• Figure 5: Turbulent steady-state solution of the periodic Burgers' equation. Left panel shows the turbulent steady-state solution of the periodic Burgers' equation with $R_e = 500$ and sinusoidal forcing \ref{['eq:BurgersPeriodicForcing']} with $k_{\mathrm{min}} = 1$, $k_{\mathrm{max}} = 128$, and randomly sampled amplitudes and phases. The right panel shows $k^2 E(t,k)$ versus $k$ to illustrate the numerically-obtained turbulent steady-state exhibits the correct $E(t,k) \sim k^{-2}$ scaling in the steady-state.