[Paper Review] On discontinuous Galerkin discretizations of second-order derivatives

Ammar Hakim, G. W. Hammett+1 more|arXiv (Cornell University)|May 22, 2014

Advanced Numerical Methods in Computational Mathematics13 references3 citations

TL;DR

This paper analyzes discontinuous Galerkin (DG) discretizations of second-order derivatives, focusing on Local DG (LDG) and Recovery DG (RDG) schemes. It shows that LDG produces non-convergent slopes in $L^2$ norm due to asymmetry in upwinding, but the error decays rapidly in time-dependent diffusion problems, preserving convergence of the solution. RDG provides symmetric, higher-order convergent approximations with reduced error.

ABSTRACT

Some properties of a Local discontinuous Galerkin (LDG) algorithm are demonstrated for the problem of evaluting a second derivative $g = f_{xx}$ for a given $f$. (This is a somewhat unusual problem, but it is useful for understanding the initial transient response of an algorithm for diffusion equations.) LDG uses an auxiliary variable to break this up into two first order equations and then applies techniques by analogy to DG algorithms for advection algorithms. This introduces an asymmetry into the solution that depends on the choice of upwind directions for these two first order equations. When using piecewise linear basis functions, this LDG solution $g_h$ is shown not to converge in an $L_2$ norm because the slopes in each cell diverge. However, when LDG is used in a time-dependent diffusion problem, this error in the second derivative term is transient and rapidly decays away, so that the overall error is bounded. I.e., the LDG approximation $f_h(x,t)$ for a diffusion equation $\partial f / \partial t = f_{xx}$ converges to the proper solution (as has been shown before), even though the initial rate of change $\partial f_h / \partial t$ does not converge. We also show results from the Recovery discontinuous Galerkin (RDG) approach, which gives symmetric solutions that can have higher rates of convergence for a stencil that couples the same number of cells.

Motivation & Objective

• To analyze the convergence behavior of discontinuous Galerkin methods when computing second derivatives, particularly in the context of diffusion and elliptic problems.
• To investigate the impact of asymmetry in Local DG (LDG) schemes—induced by upwinding choices—on the accuracy of second derivative approximations.
• To compare LDG with the Recovery DG (RDG) approach, which reconstructs continuous gradients across cell interfaces to improve symmetry and convergence.
• To evaluate the implications of non-convergent second derivative approximations on time-dependent diffusion and hyperdiffusion problems.
• To assess the performance of these schemes in problems involving higher-order derivatives, such as $f_{xxxx}$, and their suitability for plasma physics and subgrid modeling.

Proposed method

• Uses piecewise linear basis functions and Legendre polynomials as test functions to derive weak forms for the second derivative operator.
• Applies the LDG method by introducing auxiliary variables $q = f_x$ and $g = q_x$, reformulating the second-order PDE as a system of first-order equations.
• Implements numerical fluxes with upwind bias to enforce stability, introducing asymmetry dependent on the choice of upwind directions.
• Employs the RDG approach by reconstructing a continuous representation of the solution across cell interfaces using least-squares fitting to compute gradients.
• Analyzes the eigenmodes of the discrete second derivative operator to identify spurious high-frequency modes that cause non-convergence in LDG.
• Performs time-dependent simulations of the diffusion equation $\partial f/\partial t = f_{xx}$ to study transient error dynamics and convergence of $f_h(x,t)$.

Experimental results

Research questions

• RQ1 Why does the LDG discretization of $f_{xx}$ fail to converge in the $L^2$ norm for piecewise linear basis functions, despite the solution converging in time-dependent problems?
• RQ2 How does the choice of upwinding direction in LDG affect the symmetry and convergence properties of the second derivative approximation?
• RQ3 Can the Recovery DG (RDG) method achieve higher convergence rates and better stability than LDG for second derivative computation?
• RQ4 What is the impact of non-convergent second derivative approximations on the solution of time-dependent diffusion and hyperdiffusion equations?
• RQ5 How do the spurious eigenmodes of the discrete $\partial^2/\partial x^2$ operator affect long-term behavior in problems like the Schrödinger equation?

Key findings

• For piecewise linear basis functions, the LDG method produces a second derivative approximation $g_h$ that does not converge in the $L^2$ norm due to diverging cell slopes, despite converging cell averages.
• The non-convergent error in LDG is transient and decays rapidly in time-dependent diffusion problems, allowing the solution $f_h(x,t)$ to converge to the true solution.
• The Recovery DG (RDG) method yields symmetric, higher-order convergent approximations for both the second derivative and time-dependent diffusion problems, with smaller errors than LDG.
• In the Schrödinger equation test case, the second derivative error remains order unity but does not prevent convergence of observables like energy, due to cancellation of spurious modes.
• For hyperdiffusion operators like $f_{xxxx}$, RDG allows a 3-cell stencil with higher accuracy, while LDG requires a 5-cell stencil when applied sequentially.
• The dominant error in LDG arises from a high-frequency eigenmode associated with the $k$-th harmonic, which decays rapidly in time, explaining the transient nature of the error.