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\begin{document}

%********************************************************
\title{Numerical simulations with the Galerkin least squares finite element method for the Burgers' equation on the real line}


\criartitulo

\runningheads{}{GLS-FEM for the Burgers' equation on the real line.}

\begin{abstract}
{\bf Abstract.} In this work we present an efficient Galerkin least squares finite element scheme to simulate the Burgers' equation on the whole real line and subjected to initial conditions with compact support. The numerical simulations are performed by considering a sequence of auxiliary spatially dimensionless Dirichlet's problems parameterized by its numerical support $\tilde{K}$. Gaining advantage from the well-known convective-diffusive effects of the Burgers' equation, computations start by choosing $\tilde{K}$ so it contains the support of the initial condition and, as solution diffuses out, $\tilde{K}$ is increased appropriately. By direct comparisons between numerical and analytic solutions and its asymptotic behavior, we conclude that the proposed scheme is accurate even for large times, and it can be applied to numerically investigate properties of this and similar equations on unbounded domains.

{\bf Keywords}. Burgers' equation on the real line, Galerkin least squares finite element method, asymptotic properties.
\end{abstract}


%********************************************************
\newsec{Introduction}

Consider the viscous Burger's equation defined on the real line:
\begin{equation}
  \label{eq:burgers}
  \frac{\p u}{\p t} + bu\frac{\p u}{\p x} = \nu \frac{\p^2 u}{\p x^2},\quad (x\in\mathbb{R}, t>0),
\end{equation}
subjected to the initial condition:
\begin{equation}
  \label{eq:initial_condition}
  u(x,0) = g(x),\quad (x\in\mathbb{R}),
\end{equation}
where $b\neq 0$ is a given parameter, $\nu > 0$ is a given viscosity coefficient and $g$ is a given function with compact support on $\mathbb{R}$.

Burgers' equation is known to have appeared firstly in $1915$ in the work of Harry Bateman \cite{Bateman1915a}, but it receives its name after the Dutch physicist J.M. Burgers, who applied this equation in the understanding of turbulent fluids \cite{Burgers1974a}. This homogeneous quasilinear parabolic partial differential equation appears in the modeling of several phenomena such as shock flows, wave propagation in combustion chambers, vehicular traffic movement, acoustic transmission, etc. (see, for instance, \cite{Fletcher1982a} and the references therein). Another import characteristic of this equation is its several well known analytic solutions in bounded and unbounded domains. Therefore, this equation is already a classical test case in mathematical analysis and numerical simulations of convective-diffusive partial differential equations.

From the analytic point of view the literature is rich in discussing solutions and properties for the Burgers' equation on bounded and unbounded regions and subjected to a variety of initial and boundary conditions (see, for instance, ~\cite{Bastos2006a,Burgers1974a, Abd-el-Malek2000a, Evans2010a, Gorguis2006a, Holland1977a, Rodin1970a, Wood2006a}). Now, from the numerical simulation point of view the majority of the studies found in the literature are concerned about the Burgers' equation defined in a bounded region and subjected to Dirichlet's boundary conditions. Several numerical schemes have been applied to simulate this problem, for instance: Finite Element Methods \cite{Aksan2005a,Aksan2006a,Arminjon1981a,Caldwell1982a,Caldwell1981a,Caldwell1987a,Dogan2004a,Fletcher1983a,Hrymak1986a,Kadalbajoo2006a,Kutluay2004a,Ladeia2013a,Ozis2003a,Ozis2005b,Shao2011a,Zhang2009a}, Finite Difference Methods \cite{Caldwell1982a,Fletcher1983a,Gulsu2006a,Kutluay1999a,Mukundan2015a}, variational schemes \cite{Aksan2004a,Caldwell1985a,Ozis1996a}, spectral methods \cite{Basdevant1986a,Khater2008a}, Hardy's multiquadric method \cite{Hon1998a}, matched asymptotic expansion methods \cite{Ozis2005a}, multisymplectic box methods \cite{Tabatabaei2007a}, Homotopy Analysis Methods \cite{Inc2008a}, the quintic B-spline collocation procedure \cite{Saka2008a}, the gradient reproducing kernel particle method \cite{Hashemian2008a}, quasi-interpolation techniques \cite{Xu2011a}, uniform Haar wavelets \cite{Jiwari2015a}.

In this work we present an efficient numerical scheme based on the Galerkin least squares finite element method to simulate Burgers' equation on the real line and subjected to initial conditions with compact support. The proposed scheme explore the convective-diffusive nature of the differential equation. If for small times the convective effects are predominant demanding very fine and localized meshes, for large times diffusion takes place and the solution tends to relax demanding less refined but large meshes. We deal with it by computing the finite element discretization of a sequence of dimensionless spatially forms of the Burgers' equation on a fixed mesh and parameterized by its domain. This idea has been proved very computational efficient producing accurate results. This is supported by direct comparisons between numerical and analytic solutions and their asymptotic behavior.

In the next section we briefly discuss the analytic solution of problem \eqref{eq:burgers}-\eqref{eq:initial_condition} and its asymptotic properties. In Section~\ref{sec:finite_element_scheme} we present the proposed time and space discretization of the spatially dimensionless form of the Burgers' equation. In Section~\ref{sec:implementation_scheme} we discuss the details of the implementation scheme. Then in Section~\ref{sec:numerical_experiments} we present numerical experiments, which endorse the efficiency and accuracy of the scheme as to its potential to be applied to investigate solution properties on the real line. Finally, in Section~\ref{sec:final_considerations} we close by summarizing the principal aspects of this work.


%********************************************************
\newsec{Analytic solution}\label{sec:analytic_solution}

Here we recall the well known closed-form expression for $u(x,t)$ obtained by J. Cole and E. Hopf \cite{Cole1951a, Hopf1950a}. Introducing $\beta(x,t)$ and $\beta_{0}(x)$ by the Hopf-Cole transformation:
\begin{equation}
  \beta(x,t) := \exp\left\{-\frac{b}{2\nu}\int_0^x u(y,t)\,dy\right\},\quad \beta_0(x) := \exp\left\{-\frac{b}{2\nu}\int_0^x g(y)\,dy\right\}
\end{equation}
one obtains that $\beta$ solves the following initial value problem for the heat equation:
\begin{align}
  &\frac{\p \beta}{\p t}  = \nu \frac{\p^{2} \theta}{\p x^{2}},\quad (x \in \mathbb{R}, t>0)\\
  &\beta(x,0) = \beta_0(x),\quad (x \in \mathbb{R}),
\end{align}
whose unique bounded solution is given by Poisson's formula:
\begin{equation}
\beta(x,t) = \frac{1}{\sqrt{4 \pi\nu t}}\int_{-\infty}^\infty e^{-\frac{|x-y|^2}{4\nu t}}\beta_0(y)\,dy,\quad (x\in\mathbb{R}, t>0).
\end{equation}
Since $\displaystyle u = -\frac{2\nu}{b} \frac{\beta_{x}}{\beta}$, it follows that:
\begin{equation}\label{eq:analytic_solution}
  u(x,t) = \frac{1}{b}\frac{\int_{-\infty}^\infty \frac{x-y}{t}e^{-\frac{|x-y|^2}{4\nu t}}\beta_0(y)\,dy}{\int_{-\infty}^\infty e^{-\frac{|x-y|^2}{4\nu t}}\beta_0(y)\,dy},\quad (x\in\mathbb{R}, t>0).
\end{equation}

This also shows that problem \eqref{eq:burgers}-\eqref{eq:initial_condition} has a unique solution $\displaystyle u(\cdot,t) \in C^{0}\left([0, \infty)\right.$, $\left. L^{1}(\mathbb{R})\right)$, given by \eqref{eq:analytic_solution} above, which satisfies: $u \in C^{\infty}(\mathbb{R} \times (0, \infty))$ and $u(\cdot,t) \in C^{0}\left((0,\infty), W^{k,p}(\mathbb{R})\right)$ for every $k \geq 1$, $p \geq 1$. Here, $W^{k,p}(\mathbb{R})$ is the Sobolev space of functions in $L^{p}(\mathbb{R})$ whose $k$-th order derivatives belong to $ L^{p}(\mathbb{R}) $. Moreover, by \eqref{eq:analytic_solution} and standard heat kernel estimates one gets that: 
\begin{align}
  &\|u(\cdot,t)\|_{L^{2}(\mathbb{R})} = O(t^{-\frac{1}{4}}), &&\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R})} = O(t^{-\frac{1}{2}}), \\
  &\|u_{x}(\cdot,t)\|_{L^{2}(\mathbb{R})} = O(t^{-\frac{3}{4}}), &&\|u_{x}(\cdot,t)\|_{L^{\infty}(\mathbb{R})} = O(t^{-1}), \\
  &\|u_{xx}(\cdot,t)\|_{L^{2}(\mathbb{R})} = O(t^{-\frac{5}{4}}), &&\|u_{xx}(\cdot,t)\|_{L^{\infty}(\mathbb{R})} = O(t^{-\frac{3}{2}}),
\end{align}
and so on.

A more refined analysis in \cite{Zingano1997a} shows that the asymptotic limits:
\begin{equation}\label{eq:asymptote}
\gamma_{p} := \lim_{t \to \infty} t^{\frac{1}{2}\left(1 - \frac{1}{p}\right)}\|u(\cdot,t)\|_{L^p(\mathbb{R})}, \quad 1 \leq p \leq \infty,
\end{equation}
are well defined and have the following values. Let $m$ be the solution mass, that is:
\begin{equation}
m = \int_{-\infty}^\infty u(x,t)\,dx = \int_{-\infty}^\infty u_0(x)\,dx.
\end{equation}
For $1 < p \leq \infty$, we have:
\begin{equation}\label{eq:gamma_p_values}
\gamma_{p} = \frac{|m|}{\sqrt{4\pi\nu}} (4\nu)^{\frac{1}{2 p}}\frac{2 \nu}{b m}\left(1 - e^{-\frac{m}{2\nu}}\right) \|\mathcal{F}\|_{L^p(\mathbb{R})}
\end{equation}
with ${\cal F} \in L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$ defined by:
\begin{equation}
  \mathcal{F}(x) = \frac{e^{-x^2}}{\lambda - h \erf(x)}
\end{equation}
where $\erf(\cdot)$ is the error function:
\begin{equation}
  \erf(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-\xi^2}\,d\xi
\end{equation}
and $\lambda$, $h$ are given by:
\begin{equation}
  \lambda = \frac{1 + e^{-\frac{b m}{2\nu}}}{2},\quad h = \frac{1 - e^{-\frac{b m}{2\nu}}}{2}.
\end{equation}
When $p = 1$, the limit \eqref{eq:asymptote} is simply:
\begin{equation}\label{eq:gamma_1}
\gamma_1 = \lim_{t\to \infty} \|u(\cdot,t)\|_{L^1(\mathbb{R})} = |m|,
\end{equation}
and we further have: $\displaystyle \|u_{x}(\cdot,t)\|_{L^{1}(\mathbb{R})} = O(t^{-\frac{1}{2}})$, $\displaystyle \|u_{xx}(\cdot,t)\|_{L^{1}(\mathbb{R})} = O(t^{-1})$, and so on.

These results will be used in Section  \ref{sec:numerical_experiments} as further evidence for the accuracy of the numerical approximation scheme developed in the next two sections.

%********************************************************
\newsec{Finite element scheme}\label{sec:finite_element_scheme}

We consider the following auxiliary Dirichlet's problem:
\begin{align}
  & \frac{\p \tilde{u}}{\p t} + \frac{2b}{(l_b-l_a)}\tilde{u}\frac{\p \tilde{u}}{\p \tilde{x}} = \frac{4\nu}{(l_b-l_a)^2}\frac{\p^2 \tilde{u}}{\p \tilde{x}^2},\quad (\tilde{x}\in (-1, 1), t>0), \label{eq:burgers_dimensionless}\\
  & \tilde{u}(\tilde{x},0) = \tilde{g}(\tilde{x}),\quad (\tilde{x}\in (-1, 1)),\label{eq:burgers_dimensionless_ic}\\
  & \tilde{u}(-1,t) = \tilde{u}(1,t) = 0,\quad (t>0), \label{eq:burgers_dimensionless_bc}
\end{align}
where $\tilde{x} := 2x/(l_b-l_a) - (l_a + l_b)/(l_b-l_a)$ is the dimensionless space variable, $[l_a, l_b]$ is the reference domain, and $\tilde{g}(\tilde{x}) := g(\tilde{x}(l_b-l_a)/2 + (l_a + l_b)/2)$. From now one we will work with this space dimensionless problem and, for the sake of simplicity, we denote the domain length $l_{ab} := l_b - l_a$, and we will omit the tilde, i.e., we will denote $\tilde{x}$ simply by $x$ and $\tilde{u}$ by $u$.

Following the Rothe's method, we start by discretizing equation \eqref{eq:burgers_dimensionless} in time. To this end, we consider the following $\theta$-scheme for the time discretization of equation \eqref{eq:burgers_dimensionless}:
\begin{equation}\label{eq:burgers_discrete_in_time}
  \begin{split}
  \frac{u^{m+1} - u^{m}}{\delta_t} &=  - \frac{2b\theta}{l_{ab}} u^{m+1}\frac{\p u^{m+1}}{\p x} - \frac{2b(1 - \theta)}{l_{ab}} u^{m}\frac{\p u^{m}}{\p x} \\
  &+ \frac{4\nu\theta}{l_{ab}^2} \frac{\p^2 u^{m+1}}{\p x^2} + \frac{4\nu(1 - \theta)}{l_{ab}^2}\frac{\p^2 u^{m}}{\p x^2}
  \end{split}
\end{equation}
where $u^0 = u(x,0)$, $u^{m}$ denotes the approximation of $u(x,t_m)$, $m=1, 2, \dotsc$, $t_m = m\delta_t$, $\delta_t$ is a given time step size and $0 \leq \theta \leq 1$. For simplicity sake, from now one we denote $u^{m+1}$ by $u$ and $u^{m}$ by $u^0$.

Now, we consider the following weak formulation of the problem defined by equations \eqref{eq:burgers_discrete_in_time}, \eqref{eq:burgers_dimensionless_ic} and \eqref{eq:burgers_dimensionless_bc}: given $u^0\in H^1_0(-1, 1)$ find $u\in H^1_0(-1, 1)$ such that:
\begin{equation}\label{eq:weak_formulation}
  \begin{split}
    &\left(\varphi, u\right) + \frac{2\theta\delta_t}{l_{ab}} \left(\varphi, b u\frac{\p u}{\p x}\right) + \frac{4\theta\delta_t}{l_{ab}^2}\left(\frac{\p \varphi}{\p x}, \nu \frac{\p u}{\p x}\right) - \\
    &\left(\varphi, u^0\right) + \frac{2(1 - \theta)\delta_t}{l_{ab}}\left(\varphi, b u^0\frac{\p u^0}{\p x}\right) + \frac{4(1 - \theta)\delta_t}{l_{ab}^2}\left(\frac{\p \varphi}{\p x}, \nu \frac{\p u^0}{\p x}\right) = 0    
  \end{split}
\end{equation}
for all $\varphi\in H_0^1(-1, 1)$. Here, and as follows, $(\cdot,\cdot)$ denotes the $L^2(-1,1)$ inner product. 

Let's consider the following second order finite element triple $(\mathcal{K}, P_2(\mathcal{K}), \Sigma)$, where the cells $\mathcal{K}\subset\mathcal{T}_h$ are line segments forming a regular triangulation $\mathcal{T}_h$ of the segment $[-1, 1]$, the element shape functions $P_2(\mathcal{K}) = \{v:\mathcal{K}\to \mathbb{R}, v(x) = a_0 + a_1x + a_2x^2, a_0,a_1,a_2\in\mathbb{R}\}$ are second order polynomials, and the degrees of freedom $\Sigma$ are located at the end points of each $\mathcal{K}$ and its middle point (see, for instance, \cite{Johnson2009a, Larson2013a}). This allows us to define the finite element space:
\[
V_h := \{v\in C^0(-1, 1): v|_{\mathcal{K}}\in P_2(\mathcal{K}), \forall \mathcal{K}\in\mathcal{T}_h\} \subset H^1_0(-1, 1).
\]

Then, following the Galerkin least squares method (see, for instance, \cite{Larson2013a}), we iteratively approximate the solution of \eqref{eq:burgers_dimensionless} subjected to \eqref{eq:burgers_dimensionless_ic} and \eqref{eq:burgers_dimensionless_bc} by the solution of the following full discrete problem: given $u_h^0\in V_h$ find $u_h\in V_h$ such that:
\begin{equation}
  \label{eq:discrete_problem}
  \begin{split}
    &\left(\varphi_i, u_h\right) + \frac{2\theta\delta_t}{l_{ab}}\left(\varphi_i, b u_h\frac{\p u_h}{\p x}\right) + \frac{4\theta\delta_t}{l_{ab}^2}\left(\frac{\p \varphi_i}{\p x}, \nu \frac{\p u_h}{\p x}\right) + \\
    & \theta \delta_t s_h(\varphi_i, u_h) - \left(\varphi_i, u_h^0\right) + \frac{2(1 - \theta)\delta_t}{l_{ab}} \left(\varphi_i, b u_h^0\frac{\p u_h^0}{\p x}\right) +\\
    & \frac{4(1 - \theta)\delta_t}{l_{ab}^2} \left(\frac{\p \varphi_i}{\p x}, \nu \frac{\p u_h^0}{\p x}\right) + (1-\theta) \delta_t s_h(\varphi_i, u_h^0) = 0
  \end{split}
\end{equation}
for all $\varphi_i$, $i = 1,2,\dotsc,N$, in the basis of the finite element space $V_h$. The Galerkin least square stabilization term $s_h$ is given by:
\begin{equation}
  \label{eq:stabilization}
  s_h(\varphi, u) := \sum_{T\in \mathcal{T}_h} \delta_T\left(-\frac{4\nu}{l_{ab}^2}\frac{\p^2\varphi}{\p x^2} + \frac{2b}{l_{ab}}\varphi \frac{\p \varphi}{\p x}, -\frac{4\nu}{l_{ab}^2}\frac{\p^2 u}{\p x^2} + \frac{2b}{l_{ab}}u\frac{\p u}{\p x}\right),
\end{equation}
and the local stabilization parameter $\delta_T$ is chosen such that \cite{BRAACK2015}:
\begin{equation*}
  \delta_T = \delta_0 h_T\left(\frac{\nu}{h_T} + \|bu_h\|_{L^\infty(-1,1)}\right)^{-1},
\end{equation*}
where $\delta_0$ is a small positive constant.

At each time step, we solve the nonlinear system of equations \eqref{eq:discrete_problem} by the Newton's method. The Newton's formulation then reads: given $u_h^{0}\in V_h$ we iteratively compute approximations $u_h^{n+1}$ of $u_h$ by iterating:
\begin{subequations}\label{eq:newton_iteration}
  \begin{equation}
    \label{eq:newton_system}
    J(u_h^{(n)})\delta u^{(n)} = -F(u_h^{(n)}) 
  \end{equation}
  \begin{equation}
    u_h^{(n+1)} = u_h^{(n)} + \delta u^{(n)}
  \end{equation}
\end{subequations}
where $F(u_h^{n})$ denotes the left-hand-side of equation \eqref{eq:discrete_problem} substituting there $u_h$ by $u_h^{n}$, $\delta u^{n}$ is the Newton update, and the Jacobian matrix $J(u) = [\mathfrak{j}_{i,j}]_{i,j=0}^{N,N}$ have its elements defined by:
\begin{equation}
  \label{eq:jacobian_elements}
  \begin{split}
    \mathfrak{j}_{i,j} &:= \left(\varphi_i, \varphi_j\right) + \frac{2\theta\delta_t}{l_{ab}}\left(\varphi_i, b \varphi_j \frac{\p u}{\p x}\right) + \frac{2\theta\delta_t}{l_{ab}}\left(\varphi_i, b u \frac{\p \varphi_j}{\p x}\right) \\
    &+ \frac{4\theta\delta_t}{l_{ab}^2}\left(\frac{\p \varphi_i}{\p x}, \nu \frac{\p \varphi_j}{\p x}\right) + \theta\delta_t s_h'(\varphi_i,u;\varphi_j)
  \end{split}
\end{equation}
where $N$ counts for the number of elements in the triangulation, and
\begin{equation}
  \begin{split}
  s_h'(\varphi_i, u;\varphi_j) := \sum_{i=0}^{N-1}\delta_{T_i}&\left(-\frac{4\nu}{l_{ab}^2}\frac{\p^2\varphi_i}{\p x^2} + \frac{2b}{l_{ab}}\varphi_i \frac{\p \varphi_i}{\p x}, \right.\\
  &\left. - \frac{4\nu}{l_{ab}^2}\frac{\p^2 \varphi_j}{\p x^2} + \frac{2b}{l_{ab}}\varphi_j\frac{\p u}{\p x} + \frac{2b}{l_{ab}}u\frac{\p \varphi_j}{\p x}\right).
  \end{split}
\end{equation}

We now lead the discussion to the implementation of this Galerkin least squares finite element method (GLS-FEM) to simulate the Burgers' equation defined on the whole real line and subjected to an initial condition with compact support. 



%********************************************************
\newsec{Implementation scheme} \label{sec:implementation_scheme}

Because of the convective-diffusive nature of the Burgers' equation, very fine meshes are demanded to accurately compute the solution for small times, but as time increases the solution tend to relax allowing the application of less refined meshes. By assuming an initial condition with compact support numerical simulations of the auxiliary Dirichlet's problem \eqref{eq:burgers_dimensionless}-\eqref{eq:burgers_dimensionless_bc} may produce accurate solutions for finite times. To ensure the accuracy we just need to choose appropriate time step and spatial mesh length, and pick the domain $[l_a, l_b]$ sufficiently large. However, the larger the physical time we would like to consider the larger $[l_a, l_b]$ should be.

The convective effects are predominant for small times and it is appropriate to work with a small $[l_a, l_b]$, which reduces the demanding on the number of vertices of the finite element mesh. On the other hand, as time increases, the solution spreads out demanding a larger $[l_a, l_b]$ but a less refined mesh. We deal with this paradigm as follows.

Let $K := \overline{\{x\in\mathbb{R}:~g(x)\neq 0\}} \subset \mathbb{R}$ be the compact support of the initial condition. Without loss of generality, we assume that $0\in K$ and denote $d = \max_{x\in K} \{|x|\}$. Also, let's denote by $\mathcal{K}_\Gamma$ the boundary elements of the finite element space $(\mathcal{K}$, $P_2(\mathcal{K})$, $\Sigma)$. With this in mind, the implementation idea is to start simulating the auxiliary Dirichlet's problem \eqref{eq:burgers_dimensionless}-\eqref{eq:burgers_dimensionless_bc} by choosing an appropriate domain $[l_a, l_b]$ such that $\min\{l_a, l_b\} > d$. Then, at each time iteration $m$ we check if the numerical support $\tilde{K}_{m} := \{x\in\Sigma:~|u_h^{m}(x)| > 10^{-15}\}$ is still a subset of $\mathcal{T}_h\setminus \{\mathcal{K}_\Gamma\}$. If it is not the case, then we increase $l_a$ and/or $l_b$, interpolate the numerical solution $u_h^{m}$ onto $\mathcal{T}_{h}$, and we solve the next time step.

We point out that the above implementation scheme does not demand one to rewrite the finite element triangulation at each increasing of the reference domain $[l_a, l_b]$, since we are always simulating using the same fixed triangulation built in the domain $[-1, 1]$.

As time increases and the solution diffuses out, it may be possible to increase the time step $\delta_t$ gaining performance of the computations. This is particularly important for us, since we are also interested in investigating the accuracy of the proposed numerical scheme for large times. Therefore, we implemented a time step corrector based on an estimate of the rate of convergence of the Newton iterations. More precisely, at each time step $m$ with step length $\delta_t^{m}$, we compute the rate convergence estimation $\rho = \left(\|\delta u^n\|_2/\|\delta u^0\|_2\right)^{1/n}$, where $n$ is the number of Newton iterations required to convergence at this time step $m$, and $\|\cdot\|_2$ denotes the $l_2$ vector norm (see \cite{Kelley2003a} for more about this typical rate convergence estimation). We assume the convergence of the Newton iterations when $\|F(u_h^n)\|_2 < 10^{-10}$. Then, if $\rho > 0.1$, we increase $\delta_t$ by $10~\%$, and if $\rho < 0.05$, we decrease it by the same percentage. To ensure a good precision, we allowed the time step corrector to take effect at most at each one-hundred time steps and we set $10^{-1}$ as the larger time step allowed.

We summarize the implementation procedure as follows:
\begin{enumerate}
\item Set a uniform mesh with $N$ vertices built in the domain $[-1, 1]$.
\item Set the finite element triangulation $\mathcal{T}_h$.
\item Set an appropriate $[l_a, l_b] > d$.
\item Set the initial solution vector $u_h^0 \leftarrow [g(x_i)]_{i=0}^{2N+1}$, where $x_i$ is the abscissa of the $i$-th degree of freedom.
\item Set the present solution vector $u_h \leftarrow u_h^0$.
\item Set the time step $\delta_t$.
\item Loop over time steps:
  \begin{enumerate}
  \item Loop over Newton steps:
    \begin{enumerate}
    \item Assemble the Newton system.
    \item Solve the system.
    \item Set $u_h \leftarrow u_h + \delta u_h$.
    \end{enumerate}
  \item If $\tilde{K} \not\subset \mathcal{T}_h\setminus \{\mathcal{K}_\Gamma\}$, then:
    \begin{enumerate}
    \item Set to double $l_a$ and/or $l_b$.
    \item Interpolate $u_h$ considering the new reference domain.
    \end{enumerate}
  \item Correct time step.
  \item Set $u^0 \leftarrow u$.
  \end{enumerate}
\end{enumerate}

The numerical simulations were implemented in C++ using the \emph{deal.II} open source finite element library \cite{dealII84, BangerthHartmannKanschat2007}. We applied the UMFPACK sparse direct linear solver implemented there to compute the Newton update $\delta u^m$ from equation \eqref{eq:newton_system}. Evaluations of the analytic solution, its asymptotic behavior and post processing procedures were performed with the help of the Python-based ecosystem \emph{Scipy} and the library for floating-point arithmetic with arbitrary precision \emph{mpmath}.


%********************************************************
\newsec{Numerical experiments}\label{sec:numerical_experiments}

Here, we present some numerical experiments to discuss on the performance and accuracy of the proposed GLS-FEM applied to the Burgers' equation on the real line. We assume the following initial condition:
\begin{equation*}
  u_0(x) = \left\{
    \begin{array}{ll}
      \displaystyle \alpha\,e^{-10x^2} &, -2 \leq x \leq 2,\\
      0 &, \mbox{otherwise}
    \end{array}
\right.,
\end{equation*}
where we have chosen $\alpha\approx 0,89206$, which give us an initial solution with compact support and mass $m = \|u_0\|_{L^1(\mathbb{R})} = 0.5$.

We first discuss on the accuracy of the transient numerical solutions. Then, we explore their asymptotic behavior.

\subsection{Transient solutions}

The numerical solutions presented here were obtained by setting the initial domain $[l_a, l_b] = [-2, 2]$, initial time step $\delta_t = 10^{-4}$ and the artificial diffusion parameter $\delta_0 = 0.5$. In the following we discuss on the performance of the proposed numerical scheme for $|b|=1$ and for different diffusion parameters $\nu$ at several times $t$.

\begin{figure}
\centering %
\subfigure{\scalebox{1}{\includegraphics[height=6cm]{profile_nu_1e0}}} %
\hspace{1cm}  %
\subfigure{\scalebox{1}{\includegraphics[height=6cm]{errors_nu_1e0}}}%
\caption{Transient numerical solutions {\it versus} analytic solutions for $b=1$ and $\nu=1$. Left: solution profiles at $t=0.0$, $0.1$, $1.0$ and $5.02$. Right: relative error on the $L^2$-norm for numerical solutions with meshes of $N = 400$ and $800$.} %
\label{fig:m=0.5-b=1-nu=1e0}
\end{figure}

Lets start by assuming $b=1$ and $\nu = 1$. Figure~\ref{fig:m=0.5-b=1-nu=1e0} presents a comparison between GLS-FEM numerical solutions and analytic solutions computed at several times $t$. The left graphic shows solution profiles at $t=0.0$, $0.1$, $1.0$ and $5.02$. Dashed lines present analytic solution profiles, and lines present the computed GLS-FEM solutions with $N=400$. For the chosen parameters the Burgers' equation is diffusion dominated (Péclet number $Pe < 1$), and we can observe that the solution rapidly diffuses out.

The right graphic in Figure~\ref{fig:m=0.5-b=1-nu=1e0} presents the evolution of the relative error of the computed numerical solutions, i.e.:
\begin{equation*}
  \epsilon(t) := \frac{\|u_h(\cdot,t) - u(\cdot,t)\|_{L^2(\mathbb{R})}}{\|u(\cdot,t)\|_{L^2(\mathbb{R})}},
\end{equation*}
where $u_h$ denotes the GLS-FEM solution and $u$ denotes the analytic solution. The integrals involved on the computation of $\epsilon$ were performed by the composite Simpsons rule with $2N+1$ sub-intervals, where $N$ is the number of cells of the triangulation of $u_h$. The figure shows $\epsilon$ for meshes with $N=400$ and $800$ at several times between $0 \leq t \leq 1000$. We observe that numerical solutions with a good accuracy of $\epsilon(t) < 10^{-4}$ were obtained for all meshes.

\begin{figure}
  \centering %
  \subfigure{\scalebox{1}{\includegraphics[height=6cm]{profile_nu_1e-1}}} %
  \hspace{1cm}  %
  \subfigure{\scalebox{1}{\includegraphics[height=6cm]{errors_nu_1e-1}}}%
  \caption{Transient numerical solutions {\it versus} analytic solutions for $b=-1$ and $\nu=10^{-1}$. Left: numerical solution with $N=400$ and analytic solution profiles at $t=0.0$, $1.0$, $5.02$, $10.0$ and $50.4$. Right: relative error on the $L^2$-norm for numerical solutions with meshes of $N = 400$, $800$ and $1600$.} %
  \label{fig:m=0.5-b=-1-nu=1e-1}
\end{figure}

Now, Figure~\ref{fig:m=0.5-b=-1-nu=1e-1} presents the results found when $b=-1$ and $\nu=0.1$. In this case, the convective effects are stronger, but the solution still strong diffusive, as we can observe by the solution profiles given on the left graphic of this figure. The numerical solution has again a good accuracy  as one can see at the right graphic of this figure. Moreover, we point out that for small times $t$ the numerical scheme has a truncation error of at least the order $h^2$. However, for large times the diffusion effects are much stronger and, together with the increasing of the time step, causes the lost of such truncation order.

\begin{figure}
  \centering %
  \subfigure{\scalebox{1}{\includegraphics[height=6cm]{profile_nu_1e-2}}} %
  \hspace{1cm}  %
  \subfigure{\scalebox{1}{\includegraphics[height=6cm]{errors_nu_1e-2}}}%
  \caption{Transient numerical solutions {\it versus} analytic solutions for $b=1$ and $\nu=10^{-2}$. Left: numerical solution with $N=400$ and analytic solution profiles at $t=0.0$, $1.0$, $5.02$, $10.0$ and $50.4$. Right: relative error on the $L^2$-norm for numerical solutions with meshes of $N = 400$, $800$ and $1600$.} %
  \label{fig:m=0.5-b=1-nu=1e-2}
\end{figure}

As we decrease the diffusion coefficient we need more refined meshes to obtain such accurate results. The Figure~\ref{fig:m=0.5-b=1-nu=1e-2} shows the comparison between numerical and analytic solution for the case of $b=1$ and $\nu=10^{-2}$. Here, the convection effects are much stronger for small times and even a mesh with $N=1600$ was not enough to ensure a relative error of $10^{-4}$. Nevertheless, the results presented in this figure indicate that further refinements will produce such accurate numerical results. We will come back to this point later.

\begin{figure}
  \centering %
  \subfigure{\scalebox{1}{\includegraphics[height=6cm]{profile_nu_1e-3}}} %
  \hspace{1cm}  %
  \subfigure{\scalebox{1}{\includegraphics[height=6cm]{errors_nu_1e-3}}}%
  \caption{Transient numerical solutions {\it versus} analytic solutions for $b=-1$ and $\nu=10^{-3}$. Left: numerical solution with $N=800$ and analytic solution profiles at $t=0.0$, $1.0$, $5.02$, $10.0$ and $50.2$. Right: relative error on the $L^2$-norm for numerical solutions with meshes of $N = 400$, $800$ and $1600$.} %
  \label{fig:m=0.5-b=-1-nu=1e-3}
\end{figure}

As we may expect by further decreasing the diffusion coefficient even a mesh with $h_T \approx 10^{-3}$ yields numerical results with a relative accuracy of just $\epsilon(t) < 10^{-1}$. This is the case we can observe in Figure~\ref{fig:m=0.5-b=-1-nu=1e-3}, which presents results of simulations for the case of $b=-1$ and $\nu = 10^{-3}$. Again, the computed relative errors indicate we may obtain more accurate solutions by working with more refined meshes, but it also indicates the time discretization scheme (more precisely, the time step size) is getting more and more importance as we decrease the diffusion parameter.

Since it is very computational demanding to compute the analytic solution for smaller values of the diffusion parameter, we are not further able to globally compare the numerical  and analytic solution by accurately evaluating the relative error $\epsilon$. However, we may still investigate the accuracy of the numerical scheme by analyzing its asymptotic behavior.

\subsection{Asymptotic behavior}

As a further evidence of the accuracy of the proposed numerical scheme, we compare the asymptotic behavior of the numerical and analytic solutions by evaluating the $\gamma_p$ limits defined in equations \eqref{eq:asymptote}, \eqref{eq:gamma_p_values}, and \eqref{eq:gamma_1}. This is done by computing what we call the numerical $\gamma_p$, which we define as:
\begin{equation}  \label{eq:numeric_gamma_p}
  \tilde{\gamma}_p := t_f^{\frac{1}{2}\left(1 - \frac{1}{p}\right)}\|u_h(\cdot,t_f)\|_{L^p(\mathbb{R})},
\end{equation}
where $t_f$ is such that $\|F(u_h(\cdot,t_f))\|_{2} < 10^{-9}$, i.e, the $l_2$ vector norm of the residual at $t=t_f$ is less or equal to $10^{-9}$. We note that this provides an approximation of $\gamma_p$ as the numerical solution $u_h$ approximates the analytic solution $u$, and $t_f$ is large enough. Moreover we observe that this is a very fine test of the accuracy of the numerical solution, since it allowed us to check it for very large times.

\begin{table}
  \centering
  \caption{Numerical $\tilde{\gamma}_p$ {\it versus} analytic $\gamma_p$ for $p=1$ and $p=2$.}
  \begin{tabular}{l|rc|cc|cc}
    $\nu$ & $N$ & $t_f$ & $\tilde{\gamma}_1$ & $\gamma_1$ & $\tilde{\gamma}_2$ & $\gamma_2$ \\ \hline
    \multirow{2}{*}{$1.0$}    & $400$ & $2305.52$ & $0.500000$ & \multirow{2}{*}{$0.500000$} & $0.223280$ & \multirow{2}{*}{$0.223280$} \\
                              & $800$ & $2255.52$ & $0.500000$ &                             & $0.223280$ &  \\ \hline
    \multirow{3}{*}{$10^{-1}$} & $400$ & $5944.52$ & $0.499996$ & \multirow{3}{*}{$0.500000$} & $0.392037$ & \multirow{3}{*}{$0.392044$} \\
                              & $800$ & $5251.52$ & $0.499999$ &                             & $0.392039$ &  \\
                             & $1600$ & $4505.52$ & $0.500000$ &                             & $0.392039$ &  \\ \hline
    \multirow{3}{*}{$10^{-2}$} & $400$ & $30613.5$ & $0.499566$ & \multirow{3}{*}{$0.500000$} & $0.540061$ & \multirow{3}{*}{$0.540443$} \\
                              & $800$ & $23221.5$ & $0.499888$ &                             & $0.540341$ &  \\
                             & $1600$ & $17602.5$ & $0.499972$ &                             & $0.540413$ &  \\ \hline
    \multirow{5}{*}{$10^{-3}$} & $400$ & $74255.4$ & $0.484648$ & \multirow{5}{*}{$0.500000$} & $0.558566$ & \multirow{5}{*}{$0.571942$} \\
                              & $800$ & $59962.1$ & $0.490680$ &                             & $0.563831$ &  \\
                             & $1600$ & $54858.7$ & $0.496508$ &                             & $0.568901$ &  \\
                             & $3200$ & $41851.3$ & $0.498965$ &                             & $0.571032$ &  \\
                             & $6400$ & $34148.4$ & $0.499720$ &                             & $0.571685$ &  \\ \hline
    \multirow{6}{*}{$10^{-4}$} & $400$ & $>10^5$   & $0.507806$ & \multirow{6}{*}{$0.500000$} & $0.583116$ & \multirow{6}{*}{$0.576621$} \\
                              & $800$ & $>10^5$   & $0.493608$ &                             & $0.570964$ &  \\
                              & $1600$ & $>10^5$  & $0.483509$ &                             & $0.562234$ &  \\
                              & $3200$ & $85744.4$ & $0.479172$ &                            & $0.558452$ & \\
                              & $6400$ & $68076.2$ & $0.485955$ &                            & $0.564375$ & \\
                             & $12800$ & $65319.1$ & $0.494105$ &                            & $0.571469$ & \\ \hline
  \end{tabular}
  \label{tab:numericalXanalytic_gamma_p}
\end{table}

In Table~\ref{tab:numericalXanalytic_gamma_p} we present the computed values of $\tilde{\gamma}_p$ \emph{versus} $\gamma_p$ for $p=1$ (mass) and for $p=2$ for solutions with diffusion coefficients from $1$ to $10^{-4}$. The comparison between $\tilde{\gamma}_1$ and $\gamma_1$ corroborate that the numerical scheme is not mass conserving, but the mass loss can keep low by using an sufficiently refined mesh.

In opposition to the analytic solution given by equation~\eqref{eq:analytic_solution}, we observe that $\gamma_p$ can be easily computed also for very small diffusive parameters. This allows us to globally measure the accuracy of the numerical solution even for high convective regimes. Going back to Table~\ref{tab:numericalXanalytic_gamma_p} we can note that in the case of $\nu = 10^{-4}$ the numerical solution obtained with $N=400$ has a mass gain. This is an indicative that spurious numerical oscillations have affected this solution, which can be confirmed if one plots it. Now, by more refined meshes, the Galerkin least squares term is sufficient to stabilize the numerical solutions.

Finally we note that for stable numerical solution the $\tilde{\gamma}_2$ also under-determines $\gamma_2$. This is a qualitative indication of the good behavior of the numerical solution for large times, since $t^{1/4}\|u(\cdot,t)\|_{L^2(\mathbb{R})}$ is a monotonic increasing function for large times.

%********************************************************
\newsec{Final considerantions}\label{sec:final_considerations}

In this work we have presented an efficient Galerkin least squares finite element scheme to simulate the Burgers' equation on the whole real line and subjected to initial conditions with compact support. The scheme consists in computing the finite element discretization of a sequence of dimensionless spatially forms of the Burgers' equation on a fixed triangulation and parameterized by its domain, which is chosen to contain the numerical support of the solution at each time step.

In order to investigate the accuracy of the proposed scheme, we performed direct comparisons between numerical and analytic solutions. For moderated diffusion coefficients the comparisons showed the scheme can be very accurate if one works with sufficiently refined meshes. Moreover, by analyzing asymptotic parameters of the solutions we could argue that the scheme is accurate even for very large times.

As we may expect the scheme demands more and more refined meshes as we decrease the diffusion parameter. An alternative would be to work with automatic local refined meshes. We observe that this could be particularly tricky, because one will need to be careful at each time that the reference domain is enlarged.

Finally we recall the Burgers' equation is a prototype for many scientific related problems, and the proposed numerical scheme can be extended as well to assist the study of such related problems. For instance, it can be used to provide insights to analytic studies of problems on the whole real line. 

\begin{abstract}
{\bf Resumo.} Neste trabalho, apresentamos um eficiente esquema de elementos finitos com mínimos quadrados de Galerkin para simular a equação de Burgers na reta toda, sujeita a condições iniciais com suporte compacto. As simulações numéricas foram realizadas tomando-se uma sequência de problemas auxiliares de Dirichlet adimensionais no espaço e parametrizada pelo seu suporte numérico $\tilde{K}$. Tomando vantagem dos bem conhecidos efeitos convectivos e difusivos da equação de Burgers, as computações iniciam-se escolhendo $\tilde{K}$ de forma a conter o suporte da condição inicial e, conforme a solução se difunde, $\tilde{K}$ é aumentada apropriadamente. Por comparação direta entre as soluções analítica e numérica e, pelo seus comportamentos assintóticos, concluímos que o esquema proposto é preciso mesmo para tempos grandes. Assim, este pode ser aplicado para numericamente investigar propriedades desta e de similares equações em domínios não limitados.

{\bf Palavras-chave}. Equação de Burgers na reta, método de elementos finitos com mínimos quadrados de Galerkin; propriedades assintóticas.
\end{abstract}

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