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A finite volume scheme for nonlinear degenerate parabolic equations
Marianne Bessemoulin-Chatard, Francis Filbet
TL;DR
Nonlinear degenerate parabolic equations need spatial schemes that preserve steady-states without losing accuracy when diffusion vanishes. This paper combines convection and diffusion in finite volume fluxes and uses slope limiting; its numerical results report second-order accuracy in degenerate and non-degenerate regimes and satisfying long-time behavior.
Problem
Spatial discretization can cause entropy and its dissipation to saturate, and existing equilibrium-preserving schemes may become only first-order accurate when diffusion vanishes.
Source: §1 · p. 5Method
The finite volume scheme combines convective and diffusive terms in the flux to preserve steady-states and uses slope limiting for second-order spatial accuracy, including degenerate cases.
Source: §1 · pp. 2, 5Results
Numerical experiments report second-order accuracy in non-degenerate cases and nearly preserve it in degenerate cases, alongside satisfying long-time behavior for the new schemes.
Source: §4.1 · pp. 10–11, §4.2 · p. 13Takeaways & Limitations
The scheme provides a general design for entropy-admitting nonlinear degenerate parabolic equations that targets steady-state preservation and second-order spatial accuracy.
Source: §1 · p. 2, §5 · p. 19The stated equation class assumes r′(u) ≥ 0 and an entropy structure with r′(s) = h′(s)f(s) and f(u) ≥ 0.
Source: §1 · p. 1
Abstract
from arXiv · showhide
We propose a second order finite volume scheme for nonlinear degenerate parabolic equations. For some of these models (porous media equation, drift-diffusion system for semiconductors, ...) it has been proved that the transient solution converges to a steady-state when time goes to infinity. The present scheme preserves steady-states and provides a satisfying long-time behavior. Moreover, it remains valid and second-order accurate in space even in the degenerate case. After describing the numerical scheme, we present several numerical results which confirm the high-order accuracy in various regime degenerate and non degenerate cases and underline the efficiency to preserve the large-time asymptotic.
1 Introduction
The paper targets nonlinear parabolic equations with an entropy structure, including degenerate cases where diffusion may vanish. It proposes a finite volume scheme designed to preserve steady-states and retain second-order spatial accuracy.
1 Introduction
- The scheme targets equations with r′(u) ≥ 0, including cases where diffusion degenerates, and assumes an entropy structure given by r′(s) = h′(s)f(s) with f(u) ≥ 0.This structure corresponds to an energy or entropy, more generally a Lyapunov functional.Source: §1 · p. 1
- The method combines convective and diffusive terms in one flux to preserve steady-states, then uses slope limiting for second-order accuracy even in degenerate cases.The stated aim is to improve long-time behavior while retaining spatial accuracy when diffusion vanishes.Source: §1 · pp. 2, 5
- Earlier spatial schemes can lose accuracy when diffusion vanishes, becoming only first-order accurate, while prior high-order methods address accurate transient approximation.The paper frames spatial discretization as important because spatial error can saturate entropy and its dissipation.Source: §1 · pp. 2, 5
- Entropy and entropy-dissipation analyses have established convergence toward steady-states for some models, often with exponential decay, including porous-media and semiconductor drift-diffusion models.For the semiconductor system, convergence results are reported for linear diffusion and extended to a degenerate nonlinear-diffusion model.Source: §1 · pp. 3–5
- The paper presents numerical experiments on spatial accuracy, long-time behavior, physical models, and a test case with nonlinear convection and diffusion.The introduction identifies porous media and semiconductor drift-diffusion among the physical models considered.Source: §1 · pp. 3, 5
2 Presentation of the numerical scheme
The scheme discretizes convection and diffusion together, using an interface velocity derived from the combined potential. Interface reconstructions and slope limiting provide second-order accuracy, with distinct flux choices for linear and nonlinear convection.
2 Presentation of the numerical scheme
- The scheme is presented in one spatial dimension, with a finite volume update obtained by integrating over control volumes and using interface fluxes.The paper uses forward Euler for time discretization while focusing on spatial discretization.Source: §2 · p. 5
- For linear convection, the scheme combines convection and diffusion as u times the gradient of V + h(u), treating the negative gradient as an interface velocity.Centered approximations are used for the derivatives of V and h(u).Source: §2 · p. 6
- The linear-convection scheme uses a fully upwind flux, then reconstructs interface values with a slope limiter to obtain second-order accuracy.The Van Leer limiter is used for the second-order fully upwind scheme.Source: §2 · p. 6
- For nonlinear convection and diffusion, the method uses a local Lax-Friedrichs flux and reconstructed interface values to form a second-order scheme.The construction assumes f(u) ≥ 0 and a function h satisfying h′(u)f(u) = r′(u).Source: §2 · p. 6
- The construction is straightforward to extend to Cartesian meshes, but high-order accuracy on unstructured meshes requires piecewise linear reconstructions on each triangle.The passage says first-order schemes are easy to define on unstructured grids; obtaining high order is the difficulty.Source: §2 · p. 7
3 Properties of the scheme
The scheme’s properties are assessed through equilibrium preservation and an entropy estimate. The semi-discrete and fully discrete schemes preserve nonnegativity and steady states under their stated settings.
3.1 The semi-discrete scheme
- The semi-discrete scheme preserves nonnegativity and equilibrium for nonnegative initial data.The analysis assumes a bounded domain with homogeneous Neumann boundary conditions and strictly increasing r.Source: §3.1 · p. 7
- The semi-discrete analysis also establishes an entropy estimate over time.It introduces a discrete relative entropy and entropy dissipation, and concludes the estimate by integrating between t1 and t2.Source: §3.1 · pp. 7–8, §3 · p. 7
3.2 The fully-discrete scheme
- Under a CFL condition, the fully discrete first-order scheme preserves nonnegativity and steady states.The fully discrete scheme uses forward Euler time discretization.Source: §3.2 · pp. 8–9
4 Numerical simulations
The numerical experiments assess spatial accuracy in degenerate and nondegenerate cases, then test long-time behavior across semiconductor, porous-media, Fokker–Planck, and Buckley–Leverett models. They report second-order accuracy even under degeneracy and favorable preservation of long-time behavior with the proposed schemes.
4 Numerical simulations
- The experiments use forward Euler to focus on spatial discretization, requiring a time step limited by the square of the space step; the authors note that fully implicit time discretization is also suitable for long-time behavior.They choose a small time step for spatial-accuracy tests and use an adaptive step as the CFL condition becomes less restrictive near degeneracy or equilibrium.Source: §4 · p. 9
4.1 Order of convergence
- For the nondegenerate test, FU2 is experimentally second-order accurate and has a smaller L1 error than the Scharfetter–Gummel extended scheme.Table 1 compares experimental spatial order of convergence in L1 norm.Source: §4.1 · p. 10, Table 1 · p. 11
- The degenerate test finds FU2 more accurate than the other schemes: it is almost second-order while the Scharfetter–Gummel extended scheme is only first-order.The diffusion vanishes over an interval, and the solution loses regularity during evolution.Source: Table 2 · p. 11, §4.1 · p. 11
- For the discontinuous degenerate solution, FU2 produces a less-diffused shock than the three other schemes.The comparison is made at t = 0.3.Source: §4.1 · p. 12, Figure 3 · p. 12
4.2 The drift-diffusion system for semiconductors
- In semiconductor tests, the fully upwind schemes avoid the energy and dissipation saturation seen with classical upwind and Scharfetter–Gummel variants, with decay rates reported as independent of grid size.The 1D test also reports satisfying results with few cells; the 2D test observes the same saturation pattern.Source: Figure 4 · p. 13, §4.2 · p. 13, Figure 5 · p. 13
4.3 The porous media equation
- In porous-media tests, FU2 yields exponential decay of entropy and dissipation at rate -12 in the symmetric test; for nonsymmetric data, entropy decays at rate -2 independently of m.The Scharfetter–Gummel dissipation saturates earlier than the fully upwind results, while classical upwind saturation is greater.Source: §4.3 · p. 14
4.4 Nonlinear Fokker-Planck equations for fermions and bosons
- On a 40 × 40 × 40 grid, the 3D test shows exponential decay of discrete relative entropy, dissipation, and L1 distance to equilibrium; for super-critical M = 10, the solution converges to an equilibrium with a singular part at the origin.The 3D computation uses Δt = 10−4 and T = 10.Source: §4.4 · pp. 16–17
4.5 The Buckley-Leverett equation
- For the Buckley–Leverett test, the scheme remains valid for all ε, including ε = 0, when its fully upwind flux becomes the local Lax–Friedrichs flux.The numerical results compare well with.Source: §4.5 · p. 18
5 Conclusion
The paper presents a finite volume scheme for entropy-admitting nonlinear degenerate parabolic equations, combining convection and diffusion to preserve a class of steady states. Numerical examples show second-order accuracy and exponential convergence toward approximations of equilibrium in several physical models.
5 Conclusion
- The scheme combines convective and diffusive parts, uses upwind fluxes in the linear case or local Lax-Friedrichs fluxes in the nonlinear case, and preserves a particular type of steady state.It also yields a semi-discrete entropy estimate as a first step toward proving large-time behavior.Source: §5 · p. 19
- A slope limiter provides second-order accuracy even in the degenerate case.Numerical examples demonstrate high-order accuracy.Source: §5 · p. 19
- For the porous media, semiconductor drift-diffusion, and boson and fermion nonlinear Fokker-Planck models, approximate solutions converge exponentially to approximations of equilibrium.The authors identify proving this exponential rate with a discrete entropy/entropy-dissipation estimate as future work.Source: §5 · p. 19