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Convergence of the directional diffusion splitting method

R. Drebotiy, H. Shynkarenko

Abstract

We provide the proof of convergence of the directional diffusion splitting scheme for two-dimensional parabolic and elliptic advection-diffusion-reaction problems with certain restrictions on problem data

Convergence of the directional diffusion splitting method

Abstract

We provide the proof of convergence of the directional diffusion splitting scheme for two-dimensional parabolic and elliptic advection-diffusion-reaction problems with certain restrictions on problem data

Paper Structure

This paper contains 6 sections, 3 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Advection-diffusion-reaction problem
  3. Variational formulation
  4. Semi-discretized directional diffusion splitting scheme
  5. Convergence study
  6. Conclusions

Key Result

Lemma 5.1

Consider scalar function $w: \Omega \rightarrow \mathbb{R}$: $w(x):=\|\vec{\beta}(x)\|$. If $\vec{\beta}\ne 0$, $\nabla \cdot \vec{\beta }=0$ and $\nabla \cdot \vec{ b }\ge 0$ in $\Omega$, then $w^\prime_{\vec{\beta}}\le 0$ in $\Omega$.

Figures (1)

  • Figure 5.1: Flow diagram for lemma \ref{['VectorLemma']}.

Theorems & Definitions (8)

  • Lemma 5.1
  • proof
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Remark
  • Remark