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A new second-order consistent splitting scheme for the Natural Convection equations

Zhiyong Si, Yimei Ma, Yunxia Wang

arXiv:2609.28961v1math.NA

TL;DR

Natural convection equations couple velocity, pressure, and temperature through nonlinear terms, while second-order splitting schemes face a stability challenge associated with pressure extrapolation. The paper constructs a shifted-time scheme, proves stability for k > 4, and reports stability, error estimates, and second-order temporal convergence for the selected case k = 5.

  • Problem

    Natural convection equations are challenging to solve numerically because of their strong nonlinear terms and coupling, while second-order splitting schemes face a stability issue linked to pressure extrapolation.

    Source: §1 · p. 2

  • Method

    The scheme decouples pressure from temperature and velocity and is constructed using Taylor expansions about the shifted time level t^{n+k}.

    Source: §3.1 · p. 6, §6 · p. 19

  • Results

    For k > 4 the energy argument’s coefficient condition is satisfied; with k = 5, stability and convergence properties are established, and experiments confirm expected second-order temporal convergence.

    Source: §6 · p. 19

  • Takeaways & Limitations

    Under suitable regularity assumptions, the scheme has stability estimates and error bounds in both two- and three-dimensional settings.

    Source: §6 · p. 19

    The stated stability and error estimates require regularity assumptions on the exact solutions, and their constant depends on those solutions.

    Source: §4 · p. 12

Abstract

from arXiv · show

A novel second-order consistent splitting scheme is proposed for the natural convection equations. The scheme is constructed using Taylor expansions about the time level $t^{n+k}$, where $k$ is a parameter to be determined. It is proved to be stable for all $k>4$, with the present study focusing primarily on the case $k=5$. Compared with the conventional splitting scheme based on Taylor expansions about $t^{n+1}$, the proposed scheme exhibits improved stability. By employing the Sobolev inequality and Gronwall's lemma, we rigorously establish stability of the proposed scheme and derive error estimates in both two and three spatial dimensions. Finally, several numerical experiments are presented to verify the stability and accuracy of the proposed scheme and to demonstrate its effectiveness.

1 Introduction

Natural convection couples velocity, pressure, and temperature through nonlinear equations, making numerical solution challenging. This section introduces a shifted-time second-order splitting scheme and reports stability and error analysis in 2D and 3D.

1 Introduction

  • Strong nonlinear terms and inherent coupling among velocity, pressure, and temperature make numerical solutions of natural convection equations challenging.The equations also account for body forces and heat sources.Source: §1 · pp. 1–2
  • Whether the second-order consistent splitting scheme introduced for Navier-Stokes equations is unconditionally stable remained an open problem, with second-order pressure extrapolation identified as a difficulty.The introduction positions the natural-convection scheme against this stability and pressure-treatment challenge.Source: §1 · p. 2
  • The proposed second-order consistent splitting scheme uses Taylor expansions about t^{n+k}, with k = 5 highlighted as a stability-enhancing choice.The introduction notes that increasing k expands the convergence domain while slightly increasing truncation error.Source: §1 · p. 3
  • The work establishes stability and error estimates in 2D and 3D using the Sobolev inequality and Gronwall’s lemma, and reports numerical cases evaluating the scheme.The authors describe this as the first rigorous stability and error analysis for a second-order consistent splitting scheme for natural convection equations.Source: §1 · p. 3

2 Preliminaries

The preliminaries establish notation and collect properties used in the later stability and error analysis. They include projection and pressure tools, inequalities, and discrete Gronwall lemmas.

2 Preliminaries

  • The paper uses (·, ·) for the inner product and ∥·∥ for the L2(Ω) norm, and distinguishes vectors and vector spaces by boldface.It denotes by C an arbitrary positive constant independent of discretization parameters.Source: §2 · p. 3
  • The analysis collects Sobolev and Hölder inequalities and discrete Gronwall lemmas, including a lemma identified as critical to 3D weak-stability and local-in-time error proofs.The displayed 3D lemma bounds y_n by G* under its stated assumptions.Source: §2 · pp. 4–5
  • The Leray-Helmholtz projection decomposes u as Pu + ∇Φ, with Pu orthogonal to gradients of functions q in H^1(Ω).The Stokes pressure is defined from this projection and used in the stability and error analysis.Source: §2 · p. 4

3 A New Second-Order Consistent Splitting Scheme for Natural Convection Equations And Its Stability

The paper proposes a second-order consistent splitting scheme for natural convection and proves stability under a condition on the parameter k. Its analysis covers both two and three dimensions, with the three-dimensional result restricted to a local time interval.

3 A New Second-Order Consistent Splitting Scheme for Natural Convection Equations And Its Stability

  • The section presents the scheme as second-order accurate and supplies a rigorous stability proof for both 2D and 3D.The stability condition is k > 4.Source: §3 · p. 5, §3.2 · p. 7

3.1 A new second-order scheme for natural convection equations

  • Taylor expansions about t^(n+k) yield a second-order consistent scheme that decouples pressure from the velocity and temperature terms.The scheme accounts for body forces and heat sources.Source: §3.1 · pp. 5–6
  • The scheme is initialized with values at the first two time levels, and the stated initial error estimate is bounded by Cδt.The estimate includes velocity and temperature errors and their gradients, as well as the pressure error.Source: §3.1 · p. 6

3.2 Stability Analysis

Case 1: Rigorous proof of stability for d = 2.

Case 2: Rigorous proof of stability for d = 3.

  • For d = 3, the proof proceeds substantially in the same manner as the proof for d = 2. Owing to the invalidity of (2.3), it addresses (3.14), (3.19), and (3.27).The three-dimensional stability results are restricted to a local time T ∗, satisfying the condition given in (3.33).Source: Case 2: Rigorous proof of stability… · p. 11

4 Error Estimates

For k = 5, ν = 1, and λ = 1, the section derives error estimates for the second-order scheme in two and three dimensions. The bounds have a δt^4 right-hand side, with constants that may depend on the exact solutions and other known quantities.

Error estimates

  • The error analysis sets k = 5, ν = 1, and λ = 1, and covers d = 2, 3.The theorem also requires regularity of the exact solutions.Source: §4 · p. 12
  • The constant in the estimate is independent of δt and n, but may depend on T, Ω, and the exact solutions.The section states this dependence when completing the proof.Source: §4 · pp. 12, 17

5 Numerical results

Numerical tests show diminishing errors under mesh refinement and nearly theoretically optimal L2 convergence orders across the tested viscosity coefficients. The experiments use exact solutions and synchronized spatial and temporal refinement.

5 Numerical results

  • The tests use exact solutions on [0, 1] × [0, 1]^2, with no-slip velocity boundaries and synchronized time-step and mesh refinement.The P1b-P1-P1b element pair approximates velocity, pressure, and temperature; accuracy is assessed with L2 and H1 errors for velocity and temperature and L2 error for pressure.Source: §5 · pp. 17–18
  • The convergence-order tables cover ν = 1 and ν = 0.01, and the reported L2 convergence order is nearly theoretically optimal.The convergence trend is unchanged by increasing the viscosity coefficient.Source: Table 2 · p. 18, §5 · p. 18, Table 4 · p. 19
  • Errors decrease with mesh refinement and achieve optimal convergence rates; L2 convergence orders are nearly theoretically optimal for both tested viscosity coefficients.Increasing viscosity from 0.01 to 1 generally reduces absolute errors on the same mesh, without changing the convergence trend.Source: §5 · p. 18, Table 1 · p. 18, Table 3 · p. 19

6 Conclusion

The paper constructs a second-order consistent splitting scheme using Taylor expansions about a shifted time level. Its analysis establishes a stability condition for k > 4, while numerical experiments confirm second-order temporal convergence.

6 Conclusion

  • The proposed scheme uses Taylor expansions about shifted time level t^(n+k) and is formulated for natural-convection equations with no-slip boundary conditions.The framework generalizes the conventional construction based on expansions about t^(n+1).Source: §6 · p. 19
  • The energy argument's coefficient condition holds for k > 4, and k = 5 is the smallest admissible integer; stability estimates and error bounds are derived in two and three dimensions.These results hold under suitable regularity assumptions.Source: §6 · p. 19
  • Numerical experiments agree with the theoretical results and confirm the expected second-order temporal convergence.They also demonstrate an accurate and computationally effective approach for solving the natural-convection equations.Source: §6 · p. 19
  • Future work will examine long-time and global stability, assess how k affects accuracy, stability, and computational cost, and extend the framework to higher-order schemes.Source: §6 · p. 19
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