[Paper Review] Behavior of different numerical schemes for population genetic drift problems

Minxin Chen, Chun Liu+3 more|arXiv (Cornell University)|Oct 21, 2014

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TL;DR

This paper analyzes three finite volume schemes—upwind and two central schemes—for solving the degenerate convection-dominated genetic drift equation. It shows that only the central scheme preserving flux structure avoids artificial numerical viscosity, correctly conserving both total probability and expectation, while upwind and other central schemes introduce spurious viscosity leading to incorrect steady states.

ABSTRACT

In this paper, we focus on numerical methods for the genetic drift problems, which is governed by a degenerated convection-dominated parabolic equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary points as time evolves. In order to find a \emph{complete solution} which should keep the conservation of total probability and expectation, three different schemes based on finite volume methods are used to solve the equation numerically: one is a upwind scheme, the other two are different central schemes. We observed that all the methods are stable and can keep the total probability, but have totally different long-time behaviors concerning with the conservation of expectation. We prove that any extra infinitesimal diffusion leads to a same artificial steady state. So upwind scheme does not work due to its intrinsic numerical viscosity. We find one of the central schemes introduces a numerical viscosity term too, which is beyond the common understanding in the convection-diffusion community. Careful analysis is presented to prove that the other central scheme does work. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity must be avoided.

Motivation & Objective

• To identify numerical schemes that correctly preserve both total probability and expectation in genetic drift simulations governed by a degenerate convection-dominated parabolic PDE.
• To analyze the long-term behavior of different finite volume schemes, particularly regarding artificial steady states induced by numerical viscosity.
• To prove that any extra infinitesimal diffusion leads to the same incorrect steady-state solution, highlighting the need for viscosity-free methods.
• To demonstrate that central schemes can introduce unexpected numerical viscosity, challenging conventional understanding in convection-diffusion contexts.
• To establish that only a specific flux-based central scheme maintains the complete solution structure, including Dirac singularities at boundaries.

Proposed method

• Uses three finite volume schemes: one upwind and two central difference methods, all designed to discretize the flux in the genetic drift equation.
• Employs a finite volume formulation where the flux is either split into convection and diffusion terms (schemes 1 and 2) or treated as a whole (scheme 3).
• Applies discrete maximum principle and energy estimates to prove unconditional stability for all schemes.
• Derives analytical bounds on solution decay using discrete Hölder and Poincaré-type inequalities to study long-time behavior.
• Proves that any additional infinitesimal diffusion leads to the same artificial steady state, regardless of scheme.
• Analyzes the structure of numerical viscosity terms, showing that scheme 2 introduces a second-order numerical viscosity, while scheme 1 introduces a first-order one.

Experimental results

Research questions

• RQ1 Which numerical schemes can correctly preserve both total probability and expectation in the solution of the genetic drift equation?
• RQ2 Why do upwind and certain central schemes produce artificial steady states despite being stable?
• RQ3 How does numerical viscosity—especially from central schemes—affect the long-term behavior of solutions to degenerate convection-dominated PDEs?
• RQ4 Can a central difference scheme avoid introducing numerical viscosity, and under what conditions?
• RQ5 What structural properties of the flux discretization are necessary to ensure the correct long-time behavior in problems with Dirac singularities?

Key findings

• All three schemes are unconditionally stable and conserve total probability, but only the flux-based central scheme (scheme 3) correctly preserves the expectation over time.
• The upwind scheme introduces first-order numerical viscosity, leading to an incorrect artificial steady state, despite its stability.
• One of the central schemes (scheme 2) introduces a second-order numerical viscosity term, contrary to common belief that central schemes are viscosity-free.
• Any extra infinitesimal diffusion, regardless of magnitude, leads to the same artificial steady-state solution, proving that such diffusion is fundamentally incompatible with the complete solution.
• Scheme 3, which treats the flux as a whole using central differences, avoids spurious viscosity and yields the correct long-time behavior matching the analytical solution with Dirac singularities.
• The study reveals that standard convection-diffusion numerical methods fail for this problem due to intrinsic numerical viscosity, necessitating careful method selection.