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Well-balanced convex limiting for finite element discretizations of steady convection-diffusion-reaction equations

Petr Knobloch, Dmitri Kuzmin, Abhinav Jha

Abstract

We address the numerical treatment of source terms in algebraic flux correction schemes for steady convection-diffusion-reaction (CDR) equations. The proposed algorithm constrains a continuous piecewise-linear finite element approximation using a monolithic convex limiting (MCL) strategy. Failure to discretize the convective derivatives and source terms in a compatible manner produces spurious ripples, e.g., in regions where the coefficients of the continuous problem are constant and the exact solution is linear. We cure this deficiency by incorporating source term components into the fluxes and intermediate states of the MCL procedure. The design of our new limiter is motivated by the desire to preserve simple steady-state equilibria exactly, as in well-balanced schemes for the shallow water equations. The results of our numerical experiments for two-dimensional CDR problems illustrate potential benefits of well-balanced flux limiting in the scalar case.

Well-balanced convex limiting for finite element discretizations of steady convection-diffusion-reaction equations

Abstract

We address the numerical treatment of source terms in algebraic flux correction schemes for steady convection-diffusion-reaction (CDR) equations. The proposed algorithm constrains a continuous piecewise-linear finite element approximation using a monolithic convex limiting (MCL) strategy. Failure to discretize the convective derivatives and source terms in a compatible manner produces spurious ripples, e.g., in regions where the coefficients of the continuous problem are constant and the exact solution is linear. We cure this deficiency by incorporating source term components into the fluxes and intermediate states of the MCL procedure. The design of our new limiter is motivated by the desire to preserve simple steady-state equilibria exactly, as in well-balanced schemes for the shallow water equations. The results of our numerical experiments for two-dimensional CDR problems illustrate potential benefits of well-balanced flux limiting in the scalar case.
Paper Structure (12 sections, 6 theorems, 80 equations, 9 figures, 2 tables)

This paper contains 12 sections, 6 theorems, 80 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model problem and Galerkin discretization
  3. Convex limiting for convective terms
  4. Well-balanced convex limiting
  5. Solvability and discrete maximum principle
  6. Numerical examples
  7. Interior layers
  8. Boundary layers
  9. Circular layers
  10. Circular convection
  11. Conclusions
  12. Acknowledgments

Key Result

Lemma 1

Let $X$ be a finite-dimensional Hilbert space with inner product $(\cdot,\cdot)_X$ and norm $\|\cdot\|_X^{}$. Let $T:X\to X$ be a continuous mapping and $K>0$ a real number such that $(Tx,x)_X>0$ for any $x\in X$ with $\|x\|_X^{}=K$. Then there exists $x\in X$ such that $\|x\|_X^{}\le K$ and $Tx=0$.

Figures (9)

  • Figure 1: Fictitious nodes.
  • Figure 2: Level 0 triangulations used for Grid 1 (left) and Grid 2 (right) families of computational meshes.
  • Figure 3: Interior layers, MC (left) and WMC (right) solutions, Grid 1 / level 5.
  • Figure 5: Interior layers, WMC solution, Grid 2 / level 7.
  • Figure 6: Boundary layers, MC solutions, Grid 1 / level 4, $\varepsilon=10^{-3}, 10^{-6},$$10^{-9}$ (left to right).
  • ...and 4 more figures

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • ...and 7 more