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Explaining oscillatory behavior in convection-diffusion discretization

Constantin Bacuta

TL;DR

This work addresses oscillations in discretizations of a model convection-diffusion problem with small $\varepsilon$, introducing a framework of optimal discrete infinity error control to analyze stability and accuracy. It develops bubble-based Upwinding Petrov-Galerkin (UPG) discretizations that embed upwinding diffusion via bubble test spaces, proving near-optimal error bounds in the discrete infinity norm and, in key cases, nodal exactness. The paper shows that standard SL and SPLS schemes can exhibit non-physical oscillations due to mismatches with reduced transport problems, and demonstrates how exponential and properly scaled quadratic bubbles recover accuracy at nodes and suppress spurious modes, with explicit 1D and 2D extensions. The resulting approach provides a robust pathway to designing discretizations for multi-dimensional convection-dominated problems by leveraging one-dimensional streamlines and tensor-product constructions, with practical implications for stable simulations of boundary-layer dominated flows.

Abstract

For a model convection-diffusion problem, we address the presence of oscillatory discrete solutions, and study difficulties in recovering standard approximation results for its solution. We justify the presence of non-physical oscillations and propose ways to eliminate oscillations. A new approach for error analysis that requires establishing optimal discrete infinity error as a first step is introduced and justified. We emphasize that the discretization of two dimensional convection dominated problems benefit from the efficient discretization of the corresponding one dimensional problem along each stream line. Our results are useful in building new and robust discretizations for multi-dimensional convection dominated problems.

Explaining oscillatory behavior in convection-diffusion discretization

TL;DR

This work addresses oscillations in discretizations of a model convection-diffusion problem with small , introducing a framework of optimal discrete infinity error control to analyze stability and accuracy. It develops bubble-based Upwinding Petrov-Galerkin (UPG) discretizations that embed upwinding diffusion via bubble test spaces, proving near-optimal error bounds in the discrete infinity norm and, in key cases, nodal exactness. The paper shows that standard SL and SPLS schemes can exhibit non-physical oscillations due to mismatches with reduced transport problems, and demonstrates how exponential and properly scaled quadratic bubbles recover accuracy at nodes and suppress spurious modes, with explicit 1D and 2D extensions. The resulting approach provides a robust pathway to designing discretizations for multi-dimensional convection-dominated problems by leveraging one-dimensional streamlines and tensor-product constructions, with practical implications for stable simulations of boundary-layer dominated flows.

Abstract

For a model convection-diffusion problem, we address the presence of oscillatory discrete solutions, and study difficulties in recovering standard approximation results for its solution. We justify the presence of non-physical oscillations and propose ways to eliminate oscillations. A new approach for error analysis that requires establishing optimal discrete infinity error as a first step is introduced and justified. We emphasize that the discretization of two dimensional convection dominated problems benefit from the efficient discretization of the corresponding one dimensional problem along each stream line. Our results are useful in building new and robust discretizations for multi-dimensional convection dominated problems.
Paper Structure (15 sections, 7 theorems, 105 equations, 3 figures)

This paper contains 15 sections, 7 theorems, 105 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Optimal trial norm for the convection diffusion problem
  3. Discrete optimal trial norm for the one dimensional case
  4. The SL and SPLS discretizations exhibit non-physical oscillations
  5. Standard discretization with $C^0-P^1$ test and trial spaces
  6. Approximation for the SPLS discretization
  7. The oscillatory behavior of $P^1-P^2$ SPLS discretization
  8. The Upwinding Petrov-Galerkin method with bubble type test space
  9. The importance of the discrete infinity error control
  10. The importance of convergence in discrete infinity error norm
  11. The exponential bubble UPG provides the exact solution at the nodes
  12. Properly scaled quadratic bubble UPG provides optimal approximation
  13. Two dimensional extensions of bubble UPG discretizations
  14. Understanding the oscillation spikes near the parabolic BLs
  15. Conclusions on oscillatory behavior of convection-diffusion discreretization

Key Result

Theorem 2.1

Let $\|\cdot\|_{*}$ and $\|\cdot\|_{*,h}$ be the norms on $Q$, and ${\mathcal{M}}_h$ and assume: Let $u$ be the solution of eq:var-cdr and let $u_h$ be the unique solution of the problems eq:var-cdr-h or SPLS4model-h. Then, the following error estimate holds:

Figures (3)

  • Figure 3.1: $n=84$, Left: $L^2$ proj. of $w=x-1/2$ on $\tilde{{\mathcal{M}}}_h$ Middle: $P^1-P^2$ SPLS for Reduced Problem Right: $P^1-P^2$ SPLS for the Standard Problem, $\varepsilon=10^{-6}$
  • Figure 6.1: The quadratic UPG solution, $\varepsilon=10^{-7} \\ h=1/2^7$, and the $L^2$ orthogonal projection of $1_{|_{[0, 1]}}$ on ${\mathcal{M}}_h$.
  • Figure :

Theorems & Definitions (13)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • ...and 3 more