# Lie Symmetry Preservation by Finite Difference Schemes for the Burgers Equation

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## Abstract

**:**

## 1. Introduction

## 2. Basics and Problem Formulation

#### 2.1. Background

**Definition**

**1**

**Theorem**

**1.**

**Proposition**

**2.**

## 3. Scheme Parametrization Algorithm

- Start with a numerical scheme:$$\begin{array}{ccc}\hfill N:U\subset {M}^{\diamond n}& \u27f6& \mathrm{IR}\hfill \\ \hfill \mathbf{z}& \mapsto & N(\mathbf{z})\hfill \end{array}$$$$\begin{array}{ccc}\hfill U& =& \{\mathbf{z}\in {M}^{\diamond n}\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}\varphi (\mathbf{z})=0\}\hfill \end{array}$$
- From the prolonged action of the r-dimensional symmetry group G on ${M}^{\diamond n}$:$$\begin{array}{ccc}\hfill g\xb7\mathbf{z}& =& \overline{\mathbf{z}}({\epsilon}_{1},\dots ,{\epsilon}_{r},\mathbf{z})\hfill \end{array}$$$$\begin{array}{ccc}\hfill N(g\xb7\mathbf{z})=\tilde{N}({\epsilon}_{1},\dots ,{\epsilon}_{r},\mathbf{z})& =& 0\hfill \end{array}$$
- Then for each symmetry parameters, suppose an algebraic form that depends on constant real coefficients, that is for $i=1,\dots ,r$:$$\begin{array}{ccc}\hfill {\epsilon}_{i}& =& {f}_{i}({a}_{i}^{{1}_{i}},\dots ,{a}_{i}^{{m}_{i}},\mathbf{z})\hfill \end{array}$$The transformed discrete equation has an expression that depends on those real constant coefficients:$$\begin{array}{ccc}\hfill \overline{N}({f}_{1}({a}_{1}^{{1}_{1}},\dots ,{a}_{1}^{{m}_{1}}),\dots ,{f}_{r}({a}_{r}^{{1}_{r}},\dots ,{a}_{r}^{{m}_{r}}),\mathbf{z})& =& 0\hfill \end{array}$$
- For each symmetry parameter, compute the equivariance relation:$$\begin{array}{ccc}\hfill \rho (\mathbf{z})\xb7\mathbf{z}& =& \rho (\overline{\mathbf{z}})\xb7\overline{\mathbf{z}}\hfill \end{array}$$
- Compute conditions over the constant coefficients from the order of accuracy of the scheme:$$\begin{array}{ccc}\hfill \overline{N}({f}_{1}({a}_{1}^{{1}_{1}},\dots ,{a}_{1}^{{m}_{1}}),\dots ,{f}_{r}({a}_{r}^{{1}_{r}},\dots ,{a}_{r}^{{m}_{r}}),\mathbf{z})& =& O(\mathsf{\Delta}{x}_{1}^{{d}_{1}},\dots ,\mathsf{\Delta}{x}_{p}^{{d}_{p}})\hfill \end{array}$$

## 4. Numerical Illustration: The Burgers Equation

- Spatial translation:${\mathbf{G}}_{\mathbf{1}}:(x,t,u)\u27fc(x+{\epsilon}_{1},t,u)$
- Time translation:${\mathbf{G}}_{\mathbf{2}}:(x,t,u)\u27fc(x,t+{\epsilon}_{2},u)$
- Projection:${\mathbf{G}}_{\mathbf{3}}:(x,t,u)\u27fc(\frac{x}{1-{\epsilon}_{3}t},\frac{t}{1-{\epsilon}_{3}t},(1-{\epsilon}_{3}t)u+{\epsilon}_{3}x)$
- Scale transformation:${\mathbf{G}}_{\mathbf{4}}:(x,t,u)\u27fc(x{e}^{{\epsilon}_{4}},t{e}^{2{\epsilon}_{4}},u{e}^{-{\epsilon}_{4}})$
- Galilean boost:${\mathbf{G}}_{\mathbf{5}}:(x,t,u)\u27fc(x+{\epsilon}_{5}t,t,u+{\epsilon}_{5})$

#### 4.1. Construction of Invariant Numerical Scheme

**Remark.**

**Remark.**

#### 4.1.1. Construction of the moving frames

#### 4.1.2. Order of Accuracy

**Remark.**

**Remark.**

#### 4.2. Numerical Applications

#### 4.2.1. Self-Similar Solutions

#### 4.2.2. Galilean Invariance

## 5. Conclusions

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**Figure 1.**Burgers. left: $\nu ={10}^{-2}$, $\mathsf{\Delta}t={10}^{-2}$ and $\mathsf{\Delta}x={4.10}^{-2}$, right: $\nu ={5.10}^{-3}$, $\mathsf{\Delta}t={10}^{-2}$ and $\mathsf{\Delta}x={2.10}^{-2}$. Numerical solutions for self-similar solution. $\mathrm{CFL}=1/2$, at time $t=1$.

**Figure 2.**Evolution of the frame. Straight and uniform displacement of the frame for a fixed boost velocity $\lambda $.

**Figure 3.**Burgers. left: Standard FTCS scheme, right: Invariant FTCS scheme. Galilean parameter $\lambda $ has value from 0 to 1. $\nu ={5.10}^{-2}$, $\mathrm{CFL}=0.1$ in the referential frame, $\mathsf{\Delta}x={1.10}^{-2}$ at time $t=1$.

**Figure 4.**Burgers. left: Standard FTCS scheme, right: Invariant FTCS scheme. Galilean parameter $\lambda $ has value from 0 to $0.8$. $\nu ={10}^{-2}$, $\mathrm{CFL}=1/2$ in the referential frame, $\mathsf{\Delta}x={2.10}^{-2}$ at time $t=1$.

$\lambda $ | 0. | 0.2 | 0.4 | 0.6 | 0.8 | 1. | |

Error ${L}^{2}$ | Classic | 0.00367055 | 0.125886 | 0.214256 | 0.272098 | 0.30876 | 0.327396 |

Invariant | 0.0153383 | 0.0152615 | 0.0151855 | 0.0151103 | 0.0150359 | 0.0149625 | |

Error ${L}_{\mathrm{abs}}$ | Classic | 0.00148657 | 0.0677236 | 0.123145 | 0.161583 | 0.182489 | 0.189093 |

Invariant | 0.00462526 | 0.0046253 | 0.00462535 | 0.0046254 | 0.00462544 | 0.0046255 |

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Chhay, M.; Hamdouni, A.
Lie Symmetry Preservation by Finite Difference Schemes for the Burgers Equation. *Symmetry* **2010**, *2*, 868-883.
https://doi.org/10.3390/sym2020868

**AMA Style**

Chhay M, Hamdouni A.
Lie Symmetry Preservation by Finite Difference Schemes for the Burgers Equation. *Symmetry*. 2010; 2(2):868-883.
https://doi.org/10.3390/sym2020868

**Chicago/Turabian Style**

Chhay, Marx, and Aziz Hamdouni.
2010. "Lie Symmetry Preservation by Finite Difference Schemes for the Burgers Equation" *Symmetry* 2, no. 2: 868-883.
https://doi.org/10.3390/sym2020868