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Numerical homogenization of a linear multiscale conservation law

Claude Le Bris, Frédéric Legoll, Giulia Merlini

arXiv:2609.37546v1math.NAcs.NA

TL;DR

Multiscale conservation laws are difficult to approximate because resolving their oscillations can make direct computation prohibitively costly, while homogenization of hyperbolic problems can be intricate. This paper homogenizes the viscous modified equation of a numerical scheme and solves the resulting effective equation. Its experiments find an efficient surrogate in the classical transport case, but a different limit for shear flow.

  • Problem

    Direct numerical approximation of multiscale conservation laws can require resolving microscopic oscillations at prohibitive computational cost, and homogenization is more intricate for hyperbolic problems.

    Source: §1 · p. 3

  • Method

    The authors derive and homogenize the viscous modified equation of a numerical scheme, then solve the effective equation as an economical approximation of the original scheme.

    Source: §1 · pp. 4, 6, §2 · p. 9

  • Results

    In the classical transport case, the approach is an efficient surrogate for the original scheme and hyperbolic law as ε vanishes, whereas in shear flow it differs from the original law’s limit.

    Source: §1 · pp. 8–9

  • Takeaways & Limitations

    The effective equation can provide an efficient alternative to directly applying the Lax–Friedrichs scheme in the studied settings, with discretization adjusted as needed.

    Source: §1 · pp. 6, 8

    For shear flow, the approach is not correct in the limit c_x → 0 because the regularized and original conservation laws have different homogenized limits.

    Source: §4.2.2 · pp. 31–32

Abstract

from arXiv · show

We address the numerical approximation of a scalar conservation law with a highly oscillatory flux, in the linear setting. Taking advantage of the intrinsic numerical viscosity of the scheme employed that yields a parabolic regularization of the original hyperbolic equation, we use homogenization theory to derive an effective equation which in turn we solve numerically. Our numerical experiments show the performance of the approach on the example of the Lax-Friedrichs scheme.

1 Introduction

The paper uses the numerical viscosity of a discretization to homogenize a multiscale conservation law at the scheme level. It derives an effective equation intended to approximate the original scheme more economically, and evaluates the approach using Lax–Friedrichs.

1 Introduction

  • Homogenization for hyperbolic problems becomes more intricate in two dimensions, where a scalar effective equation may not exist for general periodic velocity fields.Adding a suitably scaled viscous term yields a fully macroscopic homogenized transport equation across dimensions.Source: §1 · pp. 3–4
  • The method homogenizes a numerical scheme’s viscous modified equation, then solves the resulting effective equation to approximate the original scheme at lower computational cost.The effective equation no longer contains ε, allowing coarser time and space discretizations.Source: §1 · pp. 4, 6
  • The Lax–Friedrichs scheme provides the demonstration case because its numerical viscosity yields a parabolic modified equation under the CFL condition.The paper scales the time and space discretization parameters with ε to obtain the desired regularization regime.Source: §1 · pp. 4–5
  • The study focuses on a linear conservation law with periodic oscillatory advection and presents the Lax–Friedrichs analysis as groundwork for more general settings.The authors identify nonlinear fluxes, locally periodic media, higher-order schemes, and different scalings as future directions.Source: §1 · pp. 7, 9
  • The experiments examine fixed discretization as ε vanishes and fixed small ε as the discretization is refined, including one- and two-dimensional tests.The final one-dimensional assessment compares accuracy at a given computational cost and cost at a given accuracy against a classical finite-difference scheme.Source: §1 · pp. 8–9
  • For classical transport homogenization, the approach is an efficient surrogate for the Lax–Friedrichs scheme and the hyperbolic law as ε vanishes; for shear flow, it differs from the original law’s limit.The regularized and original laws have different homogenized limits in the shear-flow case.Source: §1 · pp. 8–9

2 Our approach

The approach derives a modified equation for the Lax–Friedrichs scheme, uses its numerical diffusion to homogenize that equation, and extends the construction from one to two dimensions.

2 Our approach

  • Taylor expansion of a conservative finite-difference scheme yields a modified PDE, which is homogenized for each scaling of the discretization parameters with the microscale.The expansion retains the first non-trivial correction and neglects higher-order terms.Source: §2 · p. 9

2.1 Modified equation of the Lax-Friedrichs scheme (1D setting)

  • Under the CFL condition, the one-dimensional modified equation is parabolic because its diffusion is positive and bounded away from zero.Its modified transport field and diffusion depend on both the discretization parameters and the microscale.Source: §2.1 · p. 13
  • The modified equation accurately approximates the Lax–Friedrichs solution as cx and ct vanish at fixed ε, but that guarantee need not hold as ε also vanishes.The chosen regime scales h and δt with ε, with h < ε and a CFL-compatible time-step scaling.Source: §2.1 · pp. 12–13

2.2 Homogenization of the modified equation

  • In one dimension, the modified equation’s elliptic cell problem is well posed, and averaging yields a homogenized linear transport equation.The well-prepared initial condition reproduces the corrector’s small-scale oscillations, a property identified as key to strong convergence.Source: §2.2 · pp. 14–15

2.3 Extension to the two-dimensional setting

  • In two dimensions, the CFL condition makes the modified-equation diffusion matrix positive definite and ensures ellipticity of the cell problem.The cell problem is well posed by a Fredholm-alternative argument; without diffusion, the corresponding first-order problem may or may not be well posed, depending on the flow.Source: §2.3 · pp. 16–17

3 Generalities on the numerical approach

This section sets out how the homogenized approximation is constructed and compared with discrete and continuous solutions. It defines the solution references, numerical steps, and error measures used in the experiments.

3 Generalities on the numerical approach

  • The comparison framework distinguishes the modified-equation solution, its homogenized approximation, the homogenized limit of the original equation, and the oscillatory solution.A fine-grid Lax–Friedrichs solution serves as the reference discrete object.Source: §3 · pp. 17–18
  • The Lax–Friedrichs-homogenized approximation combines a numerical solution of the homogenized equation with a cell-problem corrector on the fine grid.The homogenized equation is solved on a grid independent of ε, and its solution is multiplied by the corrector approximation.Source: §3 · pp. 18–20
  • The cell problem is solved with P1 finite elements on a periodic cell using 60 elements per direction; periodic boundary conditions appear to prevent significant spurious-oscillation problems.The tests found no significant improvement from using a stabilized SUPG formulation.Source: §3 · p. 19
  • Comparisons use effective velocity, relative error in local averages, and relative error in fine-scale oscillations.Solutions traveling at different effective velocities are shifted before the average or oscillation errors are evaluated.Source: §3 · pp. 20–21

4 Numerical results

The numerical results evaluate the LxF-homogenized solution’s approximation of the Lax–Friedrichs scheme across microscale and discretization regimes, distinguishing modified-equation and homogenization errors while assessing accuracy and computational cost. They report its accuracy and computational advantages at fixed accuracy and fixed grid size.

4 Numerical results

  • The error separates into scheme-to-modified-equation error, governed mainly by c_x, and modified-equation-to-homogenized-equation error, governed mainly by ε.The experiments investigate regimes in which either contribution dominates.Source: §4 · pp. 21–22

4.1 Asymptotic behavior for vanishing ε

  • At fixed c_x, error decreases approximately linearly with ε before slowing and reaching a plateau, while smaller c_x uniformly reduces error.The plateau reflects the distance between the Lax–Friedrichs scheme and its modified equation, which depends on c_x but is uniformly bounded in ε; the homogenization error is a separate contribution.Source: §4.1 · p. 23, Figure 2 · p. 23, Figure 3 · p. 24

4.2.1 One-dimensional setting

  • In one dimension, decreasing c_x reduces the error until it approaches a lower bound close to 1%, and the effective velocity converges to the harmonic average as c_x tends to zero.The error floor is due to homogenization error at fixed ε; the velocity convergence is confirmed by the tests reported for Table 1.Source: §4.2.1 · pp. 25, 27, Figure 5 · p. 26, Table 1 · p. 28

4.2.2 Two-dimensional settings

  • In two dimensions, the approximation improves as c_x decreases for test cases 1–3, but shear-flow error increases under refinement; reducing ε at fixed c_x restores convergence.For cases 2 and 3, the error falls below 10% when c_x ≤ 0.01, although their limiting transport behavior is supported by numerical results rather than theoretical guarantees. In shear flow, the error increase is consistent with noncommuting limits; the authors suggest it may depend critically on ε/c_x.Source: §4.2.2 · pp. 28, 30, 31, 32
  • The chosen scaling h = c_x ε affects the modified equation’s viscosity and homogenized limit; an alternative scaling may yield a different limit, including non-local diffusion with memory for shear flow.The paper proposes investigating different scalings as future work; for incompressible flows, h = c_x ε² is given as an alternative that may better approximate the hyperbolic regime.Source: §4.2.2 · p. 34

4.3 Performance assessment

  • The performance assessment compares a homogenized approach with a finite-difference scheme on the oscillatory equation, evaluating accuracy at equal cost and cost at equal accuracy.The reference solution is computed with a highly refined Lax–Wendroff scheme; the homogenized transport equation has an analytical solution here, though the authors retain numerical solution as a more general comparison framework.Source: §4.3 · p. 35
  • Classical finite-difference schemes must resolve the microscale with h ≪ ε, whereas the proposed approach is designed for coarser grids with h > ε.The comparison uses Lax–Friedrichs as a prototype classical scheme; the authors expect the discussed behavior to be essentially independent of that choice.Source: §4.3 · pp. 35–36
  • The methods are compared at the same grid step and computational cost against the reference U ε exact, measuring errors in local averages and oscillations.For the homogenized problem, cx is decoupled from h and fixed at 1e-4 as a regularization parameter.Source: §4.3 · p. 36
  • For h from 0.01 to 0.1 with ε = 0.01, Lax–Friedrichs has errors close to 100%, while the homogenized method’s local-average and oscillation errors are 10%–0.1% and 10%–1%.The homogenized method’s local-average error decreases at second order and its oscillation error at first order in h.Source: §4.3 · p. 36, Figure 13 · p. 37
  • At fixed accuracy, the LxF-homogenized approximation uses a space step up to three orders of magnitude larger than Lax–Friedrichs for local-average and oscillation errors.The comparisons use Eave values of 0.025, 0.05, 0.1, and 0.2, and Eosc values of 0.1, 0.2, and 0.3.Source: §4.3 · p. 37, Figure 14 · p. 38
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