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Trajectory-Based RBF Collocation Method for Surface Advection-Diffusion Equations

Xiaobin Li, Leevan Ling, Yizhong Sun

TL;DR

This paper addresses solving surface advection–diffusion equations on curved manifolds by introducing the Trajectory-Based RBF Collocation (TBRBF) method. It couples a trajectory-based characteristic ODE with a diffusion PDE (an operator-split characteristic, OSC) and derives a time-continuous embedded PDE in a narrow band using push-forward operators and an advection-aligned frame, proving equivalence to the original surface PDE. The authors provide a rigorous embedding-based equivalence (no operator-splitting error), develop a Kansa-type RBF collocation algorithm for the OSC system, and validate the approach through extensive numerical experiments across circles, tori, complex implicit surfaces, and even point clouds, showing superior stability and accuracy in advection-dominated regimes. The method offers a meshfree, efficient framework for surface PDEs with broad applicability to complex geometries and potential extensions to Navier–Stokes and thin-shell problems, while noting limitations with shocks and oscillations that remain for future work.

Abstract

We introduce a Trajectory-Based RBF Collocation (TBRBF) method for solving surface advection-diffusion equations on smooth, compact manifolds. TBRBF decouples advection and diffusion by applying a characteristic treatment with a Kansa-type RBF collocation method for diffusion PDE, which yields an operator-split characteristic (OSC) system comprising a characteristic ODE and a diffusion PDE. We rigorously prove the equivalence between the OSC system and the original surface PDE on manifolds by embedding the latter into a narrow band domain. Using an intrinsic approach, we construct a time-continuous embedded PDE with push-forward operators in each chart of the atlas and establish its equivalence with the OSC system in the narrow band. Restricting the solution back to the manifold recovers the OSC system on manifolds, ensuring that the method introduces no operator splitting error. Extensive numerical experiments confirm the robust stability and accuracy of the proposed method.

Trajectory-Based RBF Collocation Method for Surface Advection-Diffusion Equations

TL;DR

This paper addresses solving surface advection–diffusion equations on curved manifolds by introducing the Trajectory-Based RBF Collocation (TBRBF) method. It couples a trajectory-based characteristic ODE with a diffusion PDE (an operator-split characteristic, OSC) and derives a time-continuous embedded PDE in a narrow band using push-forward operators and an advection-aligned frame, proving equivalence to the original surface PDE. The authors provide a rigorous embedding-based equivalence (no operator-splitting error), develop a Kansa-type RBF collocation algorithm for the OSC system, and validate the approach through extensive numerical experiments across circles, tori, complex implicit surfaces, and even point clouds, showing superior stability and accuracy in advection-dominated regimes. The method offers a meshfree, efficient framework for surface PDEs with broad applicability to complex geometries and potential extensions to Navier–Stokes and thin-shell problems, while noting limitations with shocks and oscillations that remain for future work.

Abstract

We introduce a Trajectory-Based RBF Collocation (TBRBF) method for solving surface advection-diffusion equations on smooth, compact manifolds. TBRBF decouples advection and diffusion by applying a characteristic treatment with a Kansa-type RBF collocation method for diffusion PDE, which yields an operator-split characteristic (OSC) system comprising a characteristic ODE and a diffusion PDE. We rigorously prove the equivalence between the OSC system and the original surface PDE on manifolds by embedding the latter into a narrow band domain. Using an intrinsic approach, we construct a time-continuous embedded PDE with push-forward operators in each chart of the atlas and establish its equivalence with the OSC system in the narrow band. Restricting the solution back to the manifold recovers the OSC system on manifolds, ensuring that the method introduces no operator splitting error. Extensive numerical experiments confirm the robust stability and accuracy of the proposed method.
Paper Structure (14 sections, 3 theorems, 54 equations, 9 figures, 4 tables)

This paper contains 14 sections, 3 theorems, 54 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Embedded Advection-Diffusion Equations with Push-Forward Operators
  3. Advection-aligned orthonormal frame in each chart of the atlas and intrinsic preliminaries
  4. Time-continuous embedded PDE with push-forward operators
  5. The Trajectory-Based Method
  6. Kansa-Type RBF Collocation Algorithm for OSC System
  7. Numerical Experiments
  8. Example 1: Convergence tests on the circle
  9. Example 2: Modified Shu's linear test with non-smooth initial data
  10. Example 3: Toroidal dynamics with singularity data on the torus
  11. Example 4: Turing patterns model with advection terms
  12. Example 5: Point Clouds Extension for Turing Patterns
  13. Conclusion
  14. Implicit algebraic equations and advecton velocities for all tested manifolds

Key Result

Theorem 2.1

Let $\mathcal{M} \in \mathbb{R}^3$ be a closed $C^2$, compact, oriented, codimension 1 surface, $u: \mathcal{M} \rightarrow \mathbb{R}$ be the solution of the surface advection-diffusion equation eq:adv_diff. At each point $\mathbf p_{0}\in\mathcal{M}$, choose an atlas $A^*$ with associated chart $( solves the embedded PDE The push-forward operators are given by where $\mathbf{R}$ is defined in

Figures (9)

  • Figure 1: Schematic diagram illustrating the equivalence between the surface PDE \ref{['eq:adv_diff']} and the operator-split characteristic (OSC) system \ref{['eq:finalTB2M']}. On the left, the original surface PDE \ref{['eq:adv_diff']} and the OSC system \ref{['eq:finalTB2M']} are defined on the manifold $\mathcal{M}$, depicted as a circle. To establish the analytical equivalence, the manifold is embedded into the narrow band domain $\Omega$ on the right. In this embedding process, push-forward operators (e.g., velocity $\mathbf{v}_E$) are employed to derive the embedded PDE \ref{['eq:embedded_pde']}. After proving the equivalence of the embedded PDE \ref{['eq:embedded_pde']} and the OSC system \ref{['eq:final_system']} in $\Omega$, restrict \ref{['eq:final_system']} back to the manifold for the final result \ref{['eq:finalTB2M']}.
  • Figure 1: Construction of a local advection-aligned orthonormal frame on the surface $\mathcal{M}$ using the advection velocity as one tangent vector. The parameter domain $V$ is mapped to the surface via the atlas mapping $\Gamma=\Phi^{-1}$, where $(\theta_1,\theta_2)$ parametrize the chart $U$ on the surface $\mathcal{M}$. A third coordinate $\theta_{3}=r$ extends the chart off the surface along $\hat{N}$, producing the narrow-band mapping $X$\ref{['X-r']} from $V\times[-\delta,\delta]$ (gray rectangular area) to $\Omega_{\delta,U}$ (gray dotted area), so that each level set $r=r_c:=\text{const}$ is a chart $U_{r_c}$ on the parallel surface $\mathcal{M}_{r_c}$.
  • Figure 1: Schematic illustration of the characteristic curve $\mathcal{X}(\tau;\mathbf{x}, t)$ evolving within the narrow band domain $\Omega$ (bounded by a red dashed outline). The initial point $\mathbf{x}$ (blue dot) lies on the parallel surface $\mathcal{M}_{r_c}$ (depicted as a continuous black curve). Under the tangential velocity $\mathbf{v}_E$, $\mathbf{x}$ is carried first to $\mathcal{X}(\tau_1; \mathbf{x},t)$ (orange dot) and then further to $\mathcal{X}(\tau_2; \mathcal{X}(\tau_1; \mathbf{x}, t), \tau_1)$ (red dot). This two-stage progression is equivalent to directly moving from $\mathbf{x}$ to $\mathcal{X}(\tau_2; \mathbf{x},t)$.
  • Figure 1: (Example 2) Comparison between traditional Kansa method (top) and TBRBF method (bottom) to solve \ref{['eq:adv_diff']} on the unit circle with non-smooth initial data and different diffusion coefficients $\varepsilon \in \{1e{-3},1e{-4},1e{-5},1e{-6}\}$ ( Modified Shu's Linear test with $T = 2\pi$, $\triangle t = 0.01$, $n_P = 500$, $m = 4$).
  • Figure 2: (Example 2) Performance comparison using traditional Kansa method (top) and TBRBF method (bottom) to solve \ref{['eq:adv_diff']} on the unit circle with non-smooth initial data and varying time steps $\triangle t \in \{0.01,0.005,0.001,0.0005\}$ ( Modified Shu's Linear test with $T = 2\pi$, $n_P = 500$, $\varepsilon = 1e{-6}$, $m = 4$).
  • ...and 4 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • Remark 2.1
  • Lemma 2.2
  • Proof 1
  • Proof 2
  • Lemma 3.1
  • Proof 3