Preserving Monotonicity in Anisotropic Diffusion

Prateek Sharma; Gregory W. Hammett

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Preserving Monotonicity in Anisotropic Diffusion

Prateek Sharma, Gregory W. Hammett

arXiv:0707.2616v1Jul 18, 2007astro-phphysics.comp-phphysics.plasm-ph

TL;DR

Centered anisotropic-diffusion schemes can violate monotonicity, allowing heat to flow from lower to higher temperatures and potentially producing negative temperatures. The paper modifies symmetric and asymmetric schemes with slope limiters, which avoid this behavior but introduce perpendicular numerical diffusion, making them especially useful for astrophysical plasmas with sharp gradients.

Problem
Centered anisotropic-diffusion schemes can violate monotonicity, allowing heat to flow from lower to higher temperatures and potentially producing negative temperatures.
Method
The paper modifies symmetric and asymmetric schemes using slope limiters, with additional entropy-like limiting discussed for heat-flux control.
Results
Slope-limited methods provide physically appropriate behavior at sharp temperature extrema and avoid negative temperatures, whereas centered schemes can produce them.
Takeaways & Limitations
Limited methods are particularly useful for hot, dilute astrophysical plasmas with large temperature gradients, including disk-corona interfaces and collisionless shocks.
Takeaways & Limitations
Limited methods in their present form are not suitable for fusion applications requiring accurate perpendicular diffusion at χ∥/χ⊥∼10^9.

Abstract

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We show that standard algorithms for anisotropic diffusion based on centered differencing (including the recent symmetric algorithm) do not preserve monotonicity. In the context of anisotropic thermal conduction, this can lead to the violation of the entropy constraints of the second law of thermodynamics, causing heat to flow from regions of lower temperature to higher temperature. In regions of large temperature variations, this can cause the temperature to become negative. Test cases to illustrate this for centered asymmetric and symmetric differencing are presented. Algorithms based on slope limiters, analogous to those used in second order schemes for hyperbolic equations, are proposed to fix these problems. While centered algorithms may be good for many cases, the main advantage of limited methods is that they are guaranteed to avoid negative temperature (which can cause numerical instabilities) in the presence of large temperature gradients. In particular, limited methods will be useful to simulate hot, dilute astrophysical plasmas where conduction is anisotropic and the temperature gradients are enormous, e.g., collisionless shocks and disk-corona interface.

1. Introduction

The paper identifies monotonicity failures in centered anisotropic-conduction schemes and proposes slope-limited modifications to prevent unphysical temperature behavior. These safeguards are especially relevant when temperature gradients are large.

Motivation: Centered differencing can drive heat from lower to higher temperatures, accentuating extrema and potentially producing negative temperatures and numerical instabilities.The same issue also makes centered differencing unsuitable where anisotropic diffusion must not create new extrema.
Contribution: Both symmetric and asymmetric centered methods can be modified so temperature extrema are not accentuated.The proposed modifications limit the heat-flux contributions responsible for non-monotone behavior.
Method: Slope limiters analogous to those used in second-order hyperbolic methods limit transverse heat fluxes, while the symmetric method requires limiting both normal and transverse terms.An entropy-like limiting criterion is also discussed.
Trade-offs: Limiting introduces perpendicular numerical diffusion, so it is not suitable for laboratory plasmas requiring χ∥/χ⊥∼10^9 at modest resolution.For smooth laboratory-plasma profiles, centered differencing or higher-order finite elements may remain adequate.

2. Anisotropic thermal conduction

The paper formulates anisotropic thermal conduction on a staggered grid and compares centered asymmetric and symmetric heat-flux constructions. It highlights how interpolation and stencil choices affect diffusion, orientation sensitivity, and monotonicity.

Physical model: Anisotropic plasma conduction is primarily along magnetic field lines, with parallel and perpendicular coefficients defining heat transport relative to the local field.The heat flux divergence is added to the energy equation.
Grid and implementation: Scalars such as n, e, and T are cell-centered, while vector components such as b and q are face-centered on the staggered grid.For the symmetric method, primary heat fluxes instead reside at cell corners and are interpolated to faces.
Interpolation: Harmonic averaging of nχ is weighted toward the smaller value and preserves vanishing heat flux into a region as its density approaches zero.It is also needed for stability under the stated time-step condition.
Centered symmetric scheme: Although symmetric heat flux is self-adjoint at cell corners and has χ⊥,num independent of χ∥/χ⊥, both centered schemes can violate local heat-flow monotonicity.Such violations can produce negative temperatures for large gradients.
Centered schemes: The asymmetric scheme uses a two-point normal-gradient stencil and a six-point transverse-gradient stencil, whereas the symmetric scheme uses six-point stencils for both.The symmetric construction is therefore less sensitive to coordinate orientation relative to field lines.
Centered symmetric scheme: The symmetric method cannot diffuse a chess-board temperature pattern because averaged corner gradients vanish.This leaves the pattern undiffused under the symmetric construction.

3. Negative temperature with centered differencing

Two test problems show that centered asymmetric and symmetric differencing can produce negative temperatures under large gradients, despite physically stationary or minimum-temperature conditions.

3.1. Asymmetric method: A 2×2 hot-zone test with diagonal magnetic fields makes the asymmetric method drive heat out of an already cold grid point, producing negative temperature.The temperature later becomes positive, but the intermediate negative value can cause numerical instability.
3.1. Asymmetric method: In the diagonal-field test, the asymmetric method gives Ti,j < 0 for the first few time steps, whereas the other tested methods remain positive except entropy limiting.
3.2. Symmetric method: With an x-directed magnetic field and no temperature gradient along the field, the temperature at (i, j) should remain at 0.1.
3.2. Symmetric method: The symmetric method nevertheless produces a heat flux through a cell corner, causing the temperature at (i, j) to become negative before returning to 0.1.

4. Slope limited fluxes

Slope-limited fluxes modify transverse-gradient interpolation in centered anisotropic conduction so heat fluxes do not accentuate temperature extrema.

4.1. Limiting the asymmetric method: The asymmetric normal flux has the correct higher-to-lower-temperature sign, but its transverse term can reverse direction and requires limiting.
4. Slope limited fluxes: Both asymmetric and symmetric slope-limited methods ensure that temperature extrema are not accentuated and therefore cannot produce negative temperatures.
4.1. Limiting the asymmetric method: Slope limiters such as minmod, van Leer, and MC use neighboring values to prevent new extrema in reconstructed temperature gradients.
4.2. Limiting the symmetric method: For the symmetric method, both interpolated temperature gradients at cell corners must be limited to avoid amplifying extrema.
4.2. Limiting the symmetric method: The L2 limiter averages smoothly varying values but restricts interpolated gradients to preserve the sign and magnitude of the normal heat flux.The paper uses α = 3/4 for its results.

5. Limiting with the entropy-like source function

Entropy-like limiting enforces a nonnegative heat-gradient source condition, but transverse-term limiting alone does not guarantee monotonicity; adding slope limiting does.

Entropy-like condition: If ṡ* = −q⃗ ·∇T is positive everywhere, heat is guaranteed to flow from higher to lower temperatures.
Entropy-like condition: For the symmetric method, corner values of ṡ* are positive definite, but face interpolations can violate the condition and reverse heat flow.
Entropy-like limiting: Entropy limiting reduces transverse heat fluxes when the entropy-like condition is violated, while using L2 for the normal gradients.
Limitations: Entropy limiting alone does not guarantee that large-gradient oscillations disappear, although sufficiently small time steps can prevent late-time negative temperatures.
Combined limiting: Combining entropy limiting with slope limiting at extrema guarantees that temperature extrema are not amplified, but increases numerical diffusion.

6. Mathematical properties

The paper proves that slope-limited fluxes prevent temperature extrema from being amplified, while distinguishing local entropy-like guarantees from global conditions that may be insufficient for positivity.

Behavior at temperature extrema: Slope limiting guarantees that the maximum temperature does not increase and the minimum temperature does not decrease.At extrema, limited transverse gradients vanish, so the remaining heat flux follows the temperature gradient.
Behavior at temperature extrema: The limited normal fluxes allow heat to flow only from higher to lower temperatures, while slope limiting removes transverse heat fluxes at extrema.This prevents temperatures from decreasing at minima or increasing at maxima and avoids negative temperatures.
Behavior at temperature extrema: For minmod, van Leer, and MC limiters, the four-argument limiter vanishes unless all four arguments have the same sign.At a local temperature extremum, neighboring face-centered slopes have opposite signs or one is zero, forcing the limited transverse gradients to vanish.
The entropy-like condition: The entropy-like condition ˙s∗ = −⃗q ·⃗∇T ≥0 implies locally that heat flows from higher to lower temperatures, but its validity at selected interpolation points is not sufficient for positivity everywhere.The symmetric method can satisfy the condition at cell corners while still allowing negative cell-center temperatures; even face-pair enforcement does not cure that problem.
The entropy-like condition: With appropriate interpolation, asymmetric and slope-limited asymmetric methods satisfy a global entropy-like condition, although local slope limiting remains necessary.The paper notes that a global or pointwise entropy-like condition does not guarantee correct local heat flow everywhere.

7. Further tests

Tests of circular-field diffusion show centered methods can produce non-monotonic or negative temperatures, while limited methods avoid undershoots at the cost of greater perpendicular numerical diffusion. Convergence studies further show that most methods are roughly second-order, but their perpendicular diffusion responds differently to anisotropy.

7.1. Diffusion of a hot patch in circular magnetic field: The circular-field test initializes a hot patch with a discontinuous boundary and expects late-time uniform temperature along the magnetic ring.The domain is [−1, 1] × [−1, 1], with circular field lines and no explicit perpendicular diffusion.
7.1. Diffusion of a hot patch in circular magnetic field: At late times, all methods heat the annulus, but limited schemes show larger perpendicular numerical diffusion than centered schemes.The numerical perpendicular diffusion scales with the parallel diffusion coefficient χ for all methods in this test.
7.1. Diffusion of a hot patch in circular magnetic field: Centered asymmetric and symmetric methods develop temperature oscillations and fall below the initial minimum, whereas limited schemes avoid undershoots.The minmod limiter is much more diffusive than van Leer and MC, while entropy limiting produces only brief early undershoots.
7.1. Diffusion of a hot patch in circular magnetic field: The entropy-limited symmetric method has small early undershoots that damp quickly, reaching the initial minimum temperature 0.1 after time = 1.Slope-limited methods maintain Tmin = 0.1 at all times in the long-time test.
7.2. Convergence studies: measuring χ⊥,num: In the steady-state source-term test, perpendicular numerical diffusion increases linearly with χ∥ for all methods except the symmetric method.The symmetric method is therefore the least sensitive to increasing parallel conduction in this comparison.
7.2. Convergence studies: measuring χ⊥,num: Most methods show roughly second-order convergence of the measured perpendicular diffusion, except methods based on the minmod limiter.The ring-test estimates agree roughly with more accurate calculations from the steady-state test, with the symmetric method least diffusive.

8. Conclusions

The paper identifies negative temperatures from centered anisotropic conduction as a numerical instability risk and proposes limited methods for sharp gradients. Limited methods are most appropriate for astrophysical applications, while their perpendicular diffusion limits use in highly anisotropic fusion applications.

8. Conclusions: Centered asymmetric and symmetric differencing can drive heat from lower to higher temperatures, producing negative temperatures and imaginary sound speeds.The paper presents test problems demonstrating this behavior under large temperature gradients.
8. Conclusions: Slope-limited methods prevent negative temperatures and are intended for hot, dilute astrophysical plasmas with enormous temperature gradients.The paper specifically identifies collisionless shocks and disk-corona interfaces as motivating settings.
8. Conclusions: Limited methods are unsuitable for fusion applications requiring accurate perpendicular diffusion at χ∥/χ⊥∼109.The paper notes that limited methods remain appropriate for astrophysical applications with large temperature gradients.

Appendix A. Entropy condition for an ideal gas

The appendix relates anisotropic heat conduction to entropy production in an ideal gas and uses an entropy-like condition to enforce heat flow from higher to lower temperatures.

Appendix A. Entropy condition for an ideal gas: For an ideal gas, entropy depends on temperature and number density through S = nV k ln(T^(1/(γ−1))/n) + const.Here γ is the ratio of specific heats, equal to 5/3 for a three-dimensional mono-atomic gas.
Appendix A. Entropy condition for an ideal gas: With spatially varying density and temperature, the paper computes entropy change using the anisotropic heat flux q = −nχbb · ∇T.The entropy-rate integral is evaluated over the whole space assuming vanishing boundary contributions.
Appendix A. Entropy condition for an ideal gas: The entropy-like condition −q · ∇T ≥ 0 ensures that heat flows from higher to lower temperatures.The paper applies this condition to limit symmetric methods using face pairs.