[Paper Review] An unfitted discontinuous Galerkin scheme for conservation laws on evolving surfaces

Christian Engwer, Thomas Ranner+1 more|arXiv (Cornell University)|Feb 2, 2016

Advanced Mathematical Modeling in Engineering6 references3 citations

TL;DR

This paper presents an unfitted discontinuous Galerkin (DG) method for solving conservation laws on evolving surfaces without requiring the mesh to conform to the surface. By using a level-set representation of the surface and solving a standard transport problem on a fixed background mesh, the method achieves mass conservation and optimal convergence rates (order 1 in L¹ and L²) even for complex geometries, with numerical results confirming robustness and accuracy on shrinking circular surfaces.

ABSTRACT

Motivated by considering partial differential equations arising from conservation laws posed on evolving surfaces, a new numerical method for an advection problem is developed and simple numerical tests are performed. The method is based on an unfitted discontinuous Galerkin approach where the surface is not explicitly tracked by the mesh which means the method is extremely flexible with respect to geometry. Furthermore, the discontinuous Galerkin approach is well-suited to capture the advection driven by the evolution of the surface without the need for a space-time formulation, back-tracking trajectories or streamline diffusion. The method is illustrated by a one-dimensional example and numerical results are presented that show good convergence properties for a simple test problem.

Motivation & Objective

• To develop a robust numerical method for solving advection-dominated conservation laws on time-evolving surfaces without requiring surface-conforming meshes.
• To avoid the complexity of space-time formulations, backtracking trajectories, or streamline diffusion by reformulating the problem as a standard transport equation on an unfitted domain.
• To ensure discrete mass conservation up to machine precision in the numerical scheme.
• To demonstrate convergence and accuracy on benchmark problems involving evolving surfaces, particularly for advection driven by surface motion.
• To lay the foundation for future extensions to time-dependent problems and non-zero flux terms.

Proposed method

• The method employs an unfitted discontinuous Galerkin formulation where the surface is represented implicitly via a level-set function Φ(x,t), avoiding the need to fit the mesh to the evolving surface.
• The surface evolution is captured through a velocity field w defined via the time derivative of a diffeomorphic parametrization G(·,t), enabling the description of surface motion.
• The governing equation is reformulated as a material derivative problem: ∂•u + u∇Γ·w + ∇Γ·q = 0, with q = 0 in the presented case, reducing to pure advection due to surface motion.
• A discontinuous Galerkin variational formulation is applied on a fixed background mesh, with numerical fluxes computed at element interfaces to handle discontinuities and ensure stability.
• The method uses a consistent integration of the surface measure via the level-set function, allowing accurate quadrature on cut elements without re-meshing.
• The scheme is implemented with piecewise polynomial shape functions (k=0 in the reported tests), and mass conservation is enforced through weak enforcement of the continuity of the solution across element boundaries.

Experimental results

Research questions

• RQ1 Can an unfitted discontinuous Galerkin method achieve stable and accurate solutions for conservation laws on evolving surfaces without requiring surface-conforming meshes?
• RQ2 Does the method preserve mass to machine precision in the discrete setting, even when the surface evolves and cuts through the background mesh?
• RQ3 What convergence rates can be achieved in L¹, L², and L∞ norms for a model advection problem on a shrinking circular surface?
• RQ4 How does the method perform with non-smooth initial data, particularly in terms of numerical diffusion and sharpness of discontinuities?
• RQ5 Can the method be extended to time-dependent problems and equations with non-zero flux terms (q ≠ 0) in future work?

Key findings

• The method achieves optimal convergence order of approximately 1 in both L¹ and L² norms under h-refinement, with convergence rates approaching 1 for sufficiently fine meshes.
• For the L∞ norm, the convergence rate is less consistent at coarse resolutions but appears to approach order 1 for h < 0.01, indicating improved accuracy with mesh refinement.
• The discrete mass is conserved to machine precision: the total mass of the solution remains within 6.283185 ± 1e-6 across all refinements, closely matching the analytical value of 2π.
• The method effectively captures the advection of a constant initial concentration (u₀ ≡ 1) to a final concentration (u ≡ 2) on a shrinking circle, with minimal numerical diffusion.
• For non-constant initial data with a sharp jump (binary distribution), the method preserves the sharpness of the discontinuity reasonably well, though some numerical diffusion is observed, which can be mitigated with higher-order elements and flux limiters.
• The method remains stable and accurate even when the surface cuts through the background mesh, demonstrating robustness in handling complex, evolving geometries without re-meshing.