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Adapted Lie splitting method for convection-diffusion problems with singular convective term

Thi Tam Dang, Trung Hau Hoang, Giandomenico Orlandi

TL;DR

This work tackles convection-diffusion problems with unbounded convective coefficients by introducing an Adapted Lie splitting that decomposes the convection term via a truncated, controlled component within Lorentz spaces $L^{N,\infty}$. Framed in analytic semigroup theory, the method reformulates the problem using a modified diffusion operator $\hat{D}$ and a coefficient $\theta$, and then applies a two-flow Lie splitting to achieve first-order convergence. The authors establish a global error bound $\|u_n-u(t_n)\| \le C\tau(1+|\log\tau|)$ under suitable assumptions, with detailed local and global error analyses. Numerical experiments in two and three dimensions demonstrate stability and significant accuracy gains over the classical Lie splitting, supporting the practical viability of the proposed scheme for large-scale convection-diffusion problems with singular convective terms.

Abstract

Splitting methods are a widely used numerical scheme for solving convection-diffusion problems. However, they may lose stability in some situations, particularly when applied to convection-diffusion problems in the presence of an unbounded convective term. In this paper, we propose a new splitting method, called the "Adapted Lie splitting method", which successfully overcomes the observed instability in certain cases. Assuming that the unbounded coefficient belongs to a suitable Lorentz space, we show that the adapted Lie splitting converges to first-order under the analytic semigroup framework. Furthermore, we provide numerical experiments to illustrate our newly proposed splitting approach.

Adapted Lie splitting method for convection-diffusion problems with singular convective term

TL;DR

This work tackles convection-diffusion problems with unbounded convective coefficients by introducing an Adapted Lie splitting that decomposes the convection term via a truncated, controlled component within Lorentz spaces . Framed in analytic semigroup theory, the method reformulates the problem using a modified diffusion operator and a coefficient , and then applies a two-flow Lie splitting to achieve first-order convergence. The authors establish a global error bound under suitable assumptions, with detailed local and global error analyses. Numerical experiments in two and three dimensions demonstrate stability and significant accuracy gains over the classical Lie splitting, supporting the practical viability of the proposed scheme for large-scale convection-diffusion problems with singular convective terms.

Abstract

Splitting methods are a widely used numerical scheme for solving convection-diffusion problems. However, they may lose stability in some situations, particularly when applied to convection-diffusion problems in the presence of an unbounded convective term. In this paper, we propose a new splitting method, called the "Adapted Lie splitting method", which successfully overcomes the observed instability in certain cases. Assuming that the unbounded coefficient belongs to a suitable Lorentz space, we show that the adapted Lie splitting converges to first-order under the analytic semigroup framework. Furthermore, we provide numerical experiments to illustrate our newly proposed splitting approach.
Paper Structure (12 sections, 4 theorems, 64 equations, 3 figures, 1 algorithm)

This paper contains 12 sections, 4 theorems, 64 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary definitions and results
  3. Description of the method
  4. Model problem
  5. Adapted Lie splitting method
  6. Convergence analysis
  7. Analytical framework
  8. Local error
  9. Global error
  10. Numerical experiments in two space dimension
  11. Numerical experiments in three space dimension
  12. Conclusion and outlook

Key Result

Theorem 2.1

(Existence and uniqueness,Farroni2021FARRONI2018397) There exists a proper convex set $X_{\delta}(\Omega)$ of the space $L^{N, \infty}(\Omega)$ such that, if then the problem 1.1 admits a unique solution $u \in L^2(0,T; W^{1,2}(\Omega)) \cap C([0,T]; L^2(\Omega))$.

Figures (3)

  • Figure 1: A comparison between the Lie splitting method and the newly proposed adapted Lie splitting method when applied to convection-diffusion equation with time-independent (Figure \ref{['fig:subfig_a']}) and time-dependent (Figure \ref{['fig:subfig_b']}) boundary conditions. The error in the discrete $L^2$ norm is computed at time $T=0.1$ and is plotted by comparing the numerical solution to a reference solution. A dash-dotted slope one has been added for comparison.
  • Figure 2: A comparison between the Lie splitting method and the newly proposed adapted Lie splitting method when applied to the convection-diffusion problem \ref{['1.1']} with homogeneous boundary conditions. The error at time $T=0.1$ is computed by comparing the numerical solution to the reference solution and is plotted as a function of the time step sizes, $\tau = \frac{1}{10},\frac{1}{20},\frac{1}{40},\frac{1}{80},\frac{1}{160}$. A dash-dotted slope one is drawn for comparison. The reference solution is shown on the right picture \ref{['fig2:subfig_b']}.
  • Figure 3: A comparison between the Lie splitting method and the newly proposed adapted Lie splitting method when applied to the convection-diffusion problem \ref{['1.1']} with homogeneous boundary conditions in three dimension space. The absolute error in the discrete $L^2$ norm is computed at $T=0.1$ by comparing the numerical solution to a reference solution.

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof
  • Example 5.1
  • Example 5.2
  • Example 5.3