Discretization and convergence of the ballistic Benamou-Brenier formulation of the porous medium and Burgers equations

TL;DR

This work develops and analyzes a discretization of ballistic Benamou-Brenier reformulations for the quadratic porous medium equation and Burgers' equation, recasting evolution problems as forward-in-time global convex optimization with harmonic interpolation of density. The authors prove unconditional stability with respect to space and time discretization and establish a quadratic convergence rate for the dual solution, achieved via a proximal splitting method and a space-time FFT projection. They extend the BBB framework to multidimensional, anisotropic QPME and demonstrate robust, second-order accurate performance, including the ability to use large time steps, while addressing implementation details for efficient GPU-enabled computation. The results advance a numerically tractable, CFL-free, variational approach to non-linear PDE evolution, and point to extensions to non-periodic domains, systems of PDEs, and enhanced optimization strategies for faster convergence.

Abstract

We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as forward in time variants of the Benamou-Brenier formulation of optimal transport. This approach turns those evolution problems into global optimization problems in time and space, of which we introduce a discretization, one of whose originalities lies in the harmonic interpolation of the densities involved. We prove that the resulting schemes are unconditionally stable w.r.t. the space and time steps, and we establish a quadratic convergence rate for the dual PDE solution, under suitable assumptions. We also show that the schemes can be efficiently solved numerically using a proximal splitting method and a global space-time fast Fourier transform, and we illustrate our results with numerical experiments.

Figures (6)

• Figure 1: Top left : Barenblatt profile (\ref{['eqdef:Barenblatt']}, left), with parameter $\gamma=1$, on the time interval $[10^{-4},2\times 10^{-3}]$. Bottom left : explicit solution to Burgers' equation (\ref{['eqdef:Barenblatt']}, right), with $\nu=10^{-2}$ and $\delta = \exp(\mathrm{Re})-1$ with $\mathrm{Re}=5$ the Reynolds number. Top and bottom, center and right: numerically computed auxiliary variables $m$ and $\rho$, for the corresponding BBB formulation (\ref{['eq:pdeMRho']}).
• Figure 2: Left: log-log plot of numerical error, between the Barenblatt profile with the parameters of \ref{['fig:exactSol']}, and the numerical solution of the discretized QPME, using $4 \leq N_\tau \leq 200$ timesteps and $N_h = 5 N_\tau$ discretization points; note the second order convergence rate in the $L^1$ norm. Center: likewise for Burgers viscous equation, using $4 \leq N_\tau \leq 200$ timesteps and $N_h = 5 N_\tau$ discretization points; note the second order convergence in the $L^\infty$ norm. Right: numerical solution of the QPME with an irregular initial condition, obtained with $N_h = 128$ discretization points, and very few $N_\tau=10$ timesteps, see the discussion of large timesteps in \ref{['subsec:numQPME']}. Note that a Barenblatt-like profile is obtained at the final time, as can be expected since it is an asymptotic attractor vazquez2007porous
• Figure 3: I: Image of the Cartesian grid by the chosen diffeomorphism $\phi$. II: Visualization of the anisotropy via the ellipses $\mathcal{E}(x) := \{v \in \mathbb{R}^2 \mid \|\mathrm{D} \phi(x) v\| = r\}$, for some $r>0$. III: Numerical solution of the anisotropic QPME, on a small grid with $N_\tau=12$ and $N_h=48$. IV: Difference between the numerical and the exact solution, which is obtained as the composition of the Barenblatt profile $u^\mathrm{P}$ with the diffeomorphism $\phi$.
• Figure 4: Top: Burgers' equation with viscosity $\nu = 10^{-3}$, initial condition $u_0 = \mathbbm{1}_{0.15\leq x \leq 0.65}$, evolution time $T=0.2$, $N_\tau = 512$, $N_h=1024$, $N_{\mathop{\mathrm{prox}}\nolimits} = 6000$. Bottom: Burgers' equation with viscosity $\nu = 10^{-3}$, initial condition $u_0 = \frac{1}{2} (\mathbbm{1}_{0.15\leq x \leq 0.65} + \mathbbm{1}_{0.3\leq x \leq 0.5})$, evolution time $T=0.4$, $N_\tau = 512$, $N_h=1024$, $N_{\mathop{\mathrm{prox}}\nolimits} = 6000$.
• Figure 5: Failed attempt at the numerical solution of Burgers inviscid equation ($\nu=0$), initial condition $u_0 = \mathbbm{1}_{0.15\leq x \leq 0.65}$, evolution time $T=0.2$, $N_\tau = 256$, $N_h=512$, $N_{\mathop{\mathrm{prox}}\nolimits} = 40000$. The numerical solution is reasonably good at the final time $t=0.2$, but is incorrect, or non-converging, at earlier times. In particular, the initial condition ($t=0.0$, dark blue) is not reproduced.

Theorems & Definitions (44)

• Lemma 1.1
• Theorem 1.2
• Lemma 1.3
• Theorem 1.4
• Remark 1.5: Positivity of the density and final evolution time
• Remark 1.6: Decompositions of symmetric positive definite matrices, with integer offsets
• Proposition 2.1
• Proposition 2.3
• proof
• Lemma 2.4