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The generalized scalar auxiliary variable applied to the incompressible Boussinesq Equation

Andreas Wagner, Barbara Wohlmuth, Jan Zawallich

TL;DR

The paper develops a generalized scalar auxiliary variable (GSAV) based, second-order in time discretization for the incompressible Boussinesq equations, using a backward differentiation formula with time expansion around $t^{n+k}$ ($k\ge 3$) and an exponential integrator for the GSAV auxiliary variable to attain stability independent of the time step. It decouples temperature, velocity, and pressure via a shifted-Laplacian formulation and reformulates the scheme for $H^1$-conforming finite elements, achieving provable a priori error estimates in two dimensions. The authors provide a thorough stability and error analysis, including Grönwall-type bounds, pressure estimates, and consistency results for time derivatives and extrapolations, culminating in a tau-dependent error bound of order $\tau^4$ under suitable regularity and bootstrapping assumptions. Numerical experiments in 2D and 3D with Taylor--Hood and related discretizations validate the predicted convergence rates, demonstrate turbulence-capable Marsigli flows, and illustrate the necessity of spatial stabilization to complement GSAV-based time integration for robust simulations on large-scale problems. The work highlights a practical trade-off between the choice of $k$ (stability) and the resulting error constants, and it confirms the method’s applicability to large-scale, high-Reynolds-number flows with buoyancy effects.

Abstract

This paper introduces a second-order time discretization for solving the incompressible Boussinesq equation. It uses the generalized scalar auxiliary variable (GSAV) and a backward differentiation formula (BDF), based on a Taylor expansion around $t^{n+k}$ for $k\geq3$. An exponential time integrator is used for the auxiliary variable to ensure stability independent of the time step size. We give rigorous asymptotic error estimates of the time-stepping scheme, thereby justifying its accuracy and stability. The scheme is reformulated into one amenable to a $H^1$-conforming finite element discretization. Finally, we validate our theoretical results with numerical experiments using a Taylor--Hood-based finite element discretization and show its applicability to large-scale 3-dimensional problems.

The generalized scalar auxiliary variable applied to the incompressible Boussinesq Equation

TL;DR

The paper develops a generalized scalar auxiliary variable (GSAV) based, second-order in time discretization for the incompressible Boussinesq equations, using a backward differentiation formula with time expansion around () and an exponential integrator for the GSAV auxiliary variable to attain stability independent of the time step. It decouples temperature, velocity, and pressure via a shifted-Laplacian formulation and reformulates the scheme for -conforming finite elements, achieving provable a priori error estimates in two dimensions. The authors provide a thorough stability and error analysis, including Grönwall-type bounds, pressure estimates, and consistency results for time derivatives and extrapolations, culminating in a tau-dependent error bound of order under suitable regularity and bootstrapping assumptions. Numerical experiments in 2D and 3D with Taylor--Hood and related discretizations validate the predicted convergence rates, demonstrate turbulence-capable Marsigli flows, and illustrate the necessity of spatial stabilization to complement GSAV-based time integration for robust simulations on large-scale problems. The work highlights a practical trade-off between the choice of (stability) and the resulting error constants, and it confirms the method’s applicability to large-scale, high-Reynolds-number flows with buoyancy effects.

Abstract

This paper introduces a second-order time discretization for solving the incompressible Boussinesq equation. It uses the generalized scalar auxiliary variable (GSAV) and a backward differentiation formula (BDF), based on a Taylor expansion around for . An exponential time integrator is used for the auxiliary variable to ensure stability independent of the time step size. We give rigorous asymptotic error estimates of the time-stepping scheme, thereby justifying its accuracy and stability. The scheme is reformulated into one amenable to a -conforming finite element discretization. Finally, we validate our theoretical results with numerical experiments using a Taylor--Hood-based finite element discretization and show its applicability to large-scale 3-dimensional problems.

Paper Structure

This paper contains 17 sections, 17 theorems, 146 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Time discretization and weak formulation
  3. Assumptions and main result
  4. A priori analysis of the time integration scheme
  5. Preliminaries
  6. Stability estimates
  7. Error estimates
  8. Numerical results
  9. 2D Benchmarks
  10. Convergence rates
  11. 2D-Benchmark
  12. Stabilization in space
  13. 3D Benchmark
  14. Conclusion
  15. Appendix
  16. ...and 2 more sections

Key Result

theorem 1

Let $l\geq1$, and $k\geq 3$. Under Assumptions ass:domain, ass:constants, ass:solution, and ass:bootstrapping, we find for Scheme scheme:fem a constant $C>0$ independent of $\tau$, such that for all $m\in{\mathbb I}$ holds. Further, for sufficiently small $\tau>0$ and for all $m\in{\mathbb I}$, we have

Figures (4)

  • Figure 1: The $L^2((0,T);L^2(\Omega))$-error for real-valued $k$ and different time step sizes, and a spatial grid of $256\times 256$.
  • Figure 2: The temperature (top) and the velocity (bottom) for the Marsigli flow at $t=2,8$ with $h=\frac{1}{64}$ and $\tau=\frac{T}{2^{17}}$ illustrated via the color scale red $=1.5$ and blue $=1.0$ and particle streamlines.
  • Figure 3: Energy over time for different $k$ and $\tau=10^{-2}$.
  • Figure 4: Simulation results for the temperature with the times, going from left to right. Above the time-axis: contour plots with $\theta = 1.25$. Below the time-axis: Slice in the $x=y$ plane with $\theta$ as background color and the ${{\bf u}}$ indicated by small vectors.

Theorems & Definitions (41)

  • remark thmcounterremark
  • theorem 1
  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark
  • lemma thmcounterlemma: Ladyzhenskaya's inequality
  • remark thmcounterremark
  • ...and 31 more