1D Burgers equation

1D Burgers equation#

We consider the 1d Burgers equation

\[ \partial_t u + u \partial_x u = \nu \frac{\partial ^2u}{\partial x^2} \]

\(u_0(x) := u(x,t)\) denotes the initial condition. We choose homogeneous neumann boundary conditions in this example, i.e. $\(\partial_n u = 0, \partial \Omega\)\( with \)\Omega = (0,1)$

Time scheme#

We shall use a \(\theta\)-scheme in this case and consider the following problem

\[ \begin{align} \frac{u^{n+1}-u^n}{\Delta t} + \theta~ u^{n+1} \partial_x u^{n+1} + (1-\theta)~ u^n \partial_x u^n = \theta~\nu \frac{\partial ^2u^{n+1}}{\partial x^2} + (1-\theta)~\nu \frac{\partial ^2u^{n}}{\partial x^2} \end{align} \]

hence

\[ \begin{align} u^{n+1} + \Delta t ~ \theta~ u^{n+1} \partial_x u^{n+1} - \Delta t ~ \theta~\nu \frac{\partial ^2u^{n+1}}{\partial x^2} = u^{n} - \Delta t ~ (1-\theta)~ u^{n} \partial_x u^{n} + \Delta t ~ (1-\theta)~\nu \frac{\partial ^2u^{n}}{\partial x^2} \end{align} \]

from now on, we shall denote by \(f^n\) the right hand side of the previous equation

\[f^n := u^{n} - \Delta t ~ (1-\theta)~ u^{n} \partial_x u^{n} + \Delta t ~ (1-\theta)~\nu \frac{\partial ^2u^{n}}{\partial x^2}\]

Weak formulation#

Let \(v \in \mathcal{V}\) be a function test, we have by integrating by parts the highest order term:

\[ \begin{align} \langle v, u^{n+1} \rangle + \Delta t ~ \theta~ \langle v, u^{n+1} \partial_x u^{n+1} \rangle + \Delta t ~ \theta~\nu \langle \frac{\partial v}{\partial x}, \frac{\partial u^{n+1}}{\partial x} \rangle = \langle v, f^n \rangle \end{align} \]

The previous weak formulation is still nonlinear with respect to \(u^{n+1}\). We shall then follow the same strategy as for the previous chapter on nonlinear Poisson problem.

The strategy is to define the left hand side as a LinearForm with respect to \(v\), then linearize it around \(u^{n+1}\). We therefor can use either Picard or Newton method to treat the nonlinearity.

We consider the following linear form

\[ \begin{align} G(v;u,w) := \langle v, u \rangle + \Delta t ~ \theta~ \langle v, w \partial_x u \rangle + \Delta t ~ \theta~\nu \langle \frac{\partial v}{\partial x}, \frac{\partial u}{\partial x} \rangle , \quad \forall u,v,w \in \mathcal{V} \end{align} \]

Our problem is then

\[\begin{split} \begin{align} \mbox{Find } u^{n+1} \in \mathcal{V}, \mbox{such that}\\ G(v;u^{n+1},u^{n+1}) = l(v), \quad \forall v \in \mathcal{V} \end{align} \end{split}\]

where

\[ l(v) := \int_{\Omega} f^n v ~d\Omega, \quad \forall v \in \mathcal{V} \]

SymPDE code#

domain = Line()

V = ScalarFunctionSpace('V', domain)
u  = element_of(V, name='u')
v  = element_of(V, name='v')
w  = element_of(V, name='w')
un  = element_of(V, name='un') # time iteration
uk  = element_of(V, name='uk') # nonlinear solver iteration

x = domain.coordinates

nu = Constant('nu')
theta = Constant('theta')
dt = Constant('dt')

Defining the Linear form \(G\)#

# Linear form g: V --> R
expr = v * u + dt*theta*v*w*dx(u) + dt*theta*nu*dx(v)*dx(u)
g = LinearForm(v, integral(domain, expr))

Defining the Linear form \(l\)#

# Linear form l: V --> R
expr = v * un - dt*theta*v*un*dx(un) - dt*theta*nu*dx(v)*dx(un)
l = LinearForm(v, integral(domain, expr))

Picard Method#

\[\begin{split} \begin{align} \mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that}\\ G(v;u_{n+1},u_n) = l(v), \quad \forall v \in \mathcal{V}\_h \end{align} \end{split}\]

Picard iteration#

# Variational problem
picard = find(u, forall=v, lhs=g(v, u=u,w=uk), rhs=l(v))

Newton Method#

Let’s define

\[ F(v;u) := G(v;u,u) -l(v), \quad \forall v \in \mathcal{V} \]

Newton method writes

\[\begin{split} \begin{align} \mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that}\\ F^{\prime}(\delta u,v; u_n) = - F(v;u_n), \quad \forall v \in \mathcal{V} \\ u_{n+1} := u_{n} + \delta u, \quad \delta u \in \mathcal{V} \end{align} \end{split}\]

Newton iteration#

F = LinearForm(v, g(v,w=u)-l(v))
du  = element_of(V, name='du')

Fprime = linearize(F, u, trials=du)

# Variational problem
newton = find(du, forall=v, lhs=Fprime(du, v,u=uk), rhs=-F(v,u=uk))
Contents