A Non-compact Positivity-Preserving Scheme for Parabolic PDE via Conditional Expectation

TL;DR

The paper tackles positivity preservation for linear parabolic equations in non-divergence form with anisotropic diffusion and mixed derivatives, where classical compact schemes struggle. It introduces a Feynman–Kac–based, non-compact wide-stencil framework that discretizes four diffusion branches with positivity-preserving interpolation and adaptive branch probabilities to achieve consistency and stability under the practical scaling $\Delta t \sim h$. Boundary conditions are embedded into the update via quad-tree stopping rules and specular reflection or periodic wrapping, enabling robust treatment of Dirichlet, Neumann, and periodic boundaries without extrapolation. Theoretical results establish $L^{\infty}$ convergence rates $O(\Delta t^{1/2})$ or $O(\Delta t)$ depending on the stopping rule, with $O(\Delta t)$ under uniform stopping time and $\Delta t \sim h$; numerical experiments corroborate the rates and demonstrate boundary accuracy and positivity preservation in anisotropic diffusion settings.

Abstract

We propose a novel non-compact, positivity-preserving scheme for linear non-divergence form parabolic equations. Based on the Feynman-Kac formula, the solution is expressed as a conditional expectation of an associated diffusion process. Instead of using compact Markov chain approximations, we employ a wide stencil scheme to approximate the conditional expectation, ensuring consistency and positivity preservation. This method is effective for anisotropic diffusion with mixed derivatives, where classical schemes often fail unless the covariance matrix is diagonally dominated. A key feature of our framework is its robust treatment of boundary conditions, which avoids the accuracy loss commonly encountered in BZ and semi-Lagrangian schemes. For Dirichlet boundaries, we introduce (i) a quad-tree non-uniform stopping time scheme with O($Δt^{1/2}$) accuracy and (ii) a quad-tree uniform stopping time scheme with O($Δt$) accuracy. For Neumann boundaries, we use discrete specular reflection with O($Δt^{1/2}$) convergence, while periodic boundaries are treated using modular wrapping, achieving O($Δt$) accuracy. All analyses are conducted under the practical scaling $Δt \sim h$. Except for the uniform stopping time scheme, all schemes are explicit. The schemes are unconditionally stable and positive preserving, thanks to the probabilistic structure. To ensure consistency, a non-compact stencil is involved, which leads to the large time step constraint $Δt \sim h$. Numerical experiments confirm the predicted $L^\infty$ convergence rates for all types of boundary conditions.

Figures (4)

• Figure 1: Illustration of the above algorithm, where $\tau_1 = \tau_2 = \tau_4 = t_{n+1}$ and $\tau_3 < t_{n+1}$.
• Figure 2: Illustration of the uniform stopping time algorithm where $\tau < t_{n+1}$. Here, $\Omega^{np}$ denotes the set of grid points reached from $t_n$ with a stopping time of $t_{n+1}$.
• Figure 3: Illustration of Algorithm 3. Here, $X_4^h < x_0$ and $Y_3^h < y_0$ are reflected back into the domain.
• Figure 4: Illustration of Algorithm 4. Here, the periodic condition is applied to $X_4^h$.

Theorems & Definitions (12)

• Remark 1
• Lemma 1
• proof
• Remark 2
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof