$$ \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\x}{\boldsymbol{x}} \newcommand{\dfc}{\alpha} % diffusion coefficient \newcommand{\If}{\mathcal{I}_s} % for FEM \newcommand{\Ifb}{{I_b}} % for FEM \newcommand{\basphi}{\varphi} \newcommand{\baspsi}{\psi} \newcommand{\xno}[1]{x_{#1}} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\ds}{\, \mathrm{d}s} $$

Study guide: Time-dependent problems and variational forms

Hans Petter Langtangen [1, 2]

[1] Center for Biomedical Computing, Simula Research Laboratory
[2] Department of Informatics, University of Oslo

Oct 30, 2015

Time-dependent problems

Example: diffusion problem

 
$$ \begin{align*} \frac{\partial u}{\partial t} &= \dfc\nabla^2 u + f(\x, t),\quad &\x\in\Omega, t\in (0,T]\\ u(\x, 0) & = I(\x),\quad &\x\in\Omega\\ \frac{\partial u}{\partial n} &= 0,\quad &\x\in\partial\Omega,\ t\in (0,T] \end{align*} $$

 

A Forward Euler scheme; ideas

 
$$ \begin{equation*} [D_t^+ u = \dfc\nabla^2 u + f]^n,\quad n=1,2,\ldots,N_t-1 \end{equation*} $$

 

Solving wrt \( u^{n+1} \):

 
$$ \begin{equation*} u^{n+1} = u^n + \Delta t \left( \dfc\nabla^2 u^n + f(\x, t_n)\right) \end{equation*} $$

 

A Forward Euler scheme; stages in the discretization

 
$$ \begin{equation*} \frac{\partial \uex}{\partial t} = \dfc\nabla^2 \uex + f(\x, t) \end{equation*} $$

 

 
$$ \begin{equation*} \uex^{n+1} = \uex^n + \Delta t \left( \dfc\nabla^2 \uex^n + f(\x, t_n)\right) \end{equation*} $$

 

 
$$ \uex^n \approx u^n = \sum_{j=0}^{N} c_j^{n}\baspsi_j(\x),\quad \uex^{n+1} \approx u^{n+1} = \sum_{j=0}^{N} c_j^{n+1}\baspsi_j(\x) $$

 

 
$$ R = u^{n+1} - u^n - \Delta t \left( \dfc\nabla^2 u^n + f(\x, t_n)\right)$$

 

A Forward Euler scheme; weighted residual (or Galerkin) principle

 
$$ R = u^{n+1} - u^n - \Delta t \left( \dfc\nabla^2 u^n + f(\x, t_n)\right)$$

 

The weighted residual principle:

 
$$ \int_\Omega Rw\dx = 0,\quad \forall w\in W$$

 

results in

 
$$ \int_\Omega \left\lbrack u^{n+1} - u^n - \Delta t \left( \dfc\nabla^2 u^n + f(\x, t_n)\right) \right\rbrack w \dx =0, \quad \forall w \in W $$

 

Galerkin: \( W=V \), \( w=v \)

A Forward Euler scheme; integration by parts

Isolating the unknown \( u^{n+1} \) on the left-hand side:

 
$$ \int_{\Omega} u^{n+1}v\dx = \int_{\Omega} \left\lbrack u^n + \Delta t \left( \dfc\nabla^2 u^n + f(\x, t_n)\right) \right\rbrack v\dx $$

 

Integration by parts of \( \int\dfc(\nabla^2 u^n) v\dx \):

 
$$ \int_{\Omega}\dfc(\nabla^2 u^n)v \dx = -\int_{\Omega}\dfc\nabla u^n\cdot\nabla v\dx + \underbrace{\int_{\partial\Omega}\dfc\frac{\partial u^n}{\partial n}v \dx}_{=0\quad\Leftarrow\quad\partial u^n/\partial n=0} $$

 

Variational form:

 
$$ \begin{equation*} \int_{\Omega} u^{n+1} v\dx = \int_{\Omega} u^n v\dx - \Delta t \int_{\Omega}\dfc\nabla u^n\cdot\nabla v\dx + \Delta t\int_{\Omega}f^n v\dx,\quad\forall v\in V \end{equation*} $$

 

New notation for the solution at the most recent time levels

 
$$ \begin{equation*} \int_{\Omega} u v\dx = \int_{\Omega} u_1 v\dx - \Delta t \int_{\Omega}\dfc\nabla u_1\cdot\nabla v\dx + \Delta t\int_{\Omega}f^n v\dx \end{equation*} $$

 

or shorter

 
$$ \begin{equation*} (u,v) = (u_1,v) - \Delta t (\dfc\nabla u_1,\nabla v) + \Delta t (f^n, v) \end{equation*} $$

 

Deriving the linear systems

Insert these in

 
$$ (u, \baspsi_i) = (u_1,\baspsi_i) - \Delta t (\dfc\nabla u_1,\nabla\baspsi_i) + \Delta t (f^n,\baspsi_i) $$

 

and order terms as matrix-vector products (\( i=0,\ldots,N \)):

 
$$ \begin{equation*} \sum_{j=0}^{N} \underbrace{(\baspsi_i,\baspsi_j)}_{M_{i,j}} c_j = \sum_{j=0}^{N} \underbrace{(\baspsi_i,\baspsi_j)}_{M_{i,j}} c_{1,j} -\Delta t \sum_{j=0}^{N} \underbrace{(\nabla\baspsi_i,\dfc\nabla\baspsi_j)}_{K_{i,j}} c_{1,j} + \Delta t (f^n,\baspsi_i) \end{equation*} $$

 

Structure of the linear systems

 
$$ \begin{equation*} Mc = Mc_1 - \Delta t Kc_1 +\Delta t f \end{equation*} $$

 

 
$$ \begin{align*} M &= \{M_{i,j}\},\quad M_{i,j}=(\baspsi_i,\baspsi_j),\quad i,j\in\If\\ K &= \{K_{i,j}\},\quad K_{i,j}=(\nabla\baspsi_i,\dfc\nabla\baspsi_j),\quad i,j\in\If\\ f &= \{(f(\x,t_n),\baspsi_i)\}_{i\in\If}\\ c &= \{c_i\}_{i\in\If}\\ c_1 &= \{c_{1,i}\}_{i\in\If} \end{align*} $$

 

Computational algorithm

  1. Compute \( M \) and \( K \).
  2. Initialize \( u^0 \) by either interpolation or projection
  3. For \( n=1,2,\ldots,N_t \):

    1. compute \( b = Mc_1 - \Delta t Kc_1 + \Delta t f \)
    2. solve \( Mc = b \)
    3. set \( c_1 = c \)

Initial condition:

Example using sinusoidal basis functions

 
$$ \begin{align} \frac{\partial u}{\partial t} &= \dfc\frac{\partial^2 u}{\partial x^2},\quad &x\in (0,L),\ t\in (0,T], \tag{1}\\ u(x, 0) & = A\cos(\pi\x/L) + B\cos(10\pi x/L),\quad &x\in[0,L], \tag{2}\\ \frac{\partial u}{\partial x} &= 0,\quad &x=0,L,\ t\in (0,T] \tag{3} \tp \end{align} $$

 

 
$$ \baspsi_i = \cos(i\pi x/L)\tp$$

 

Approximating the initial condition

\( I(x)\in V \) implies perfect approximation of the initial condition:

 
$$ c_{1,1}=A,\quad c_{1,10}=B,$$

 
while \( c_{1,i}=0 \) for \( i\neq 1,10 \).

Computing the \( M \) and \( K \) matrices

Note that \( \baspsi_i \) and \( \baspsi_i' \) are orthogonal on \( [0,L] \) such that we only need to compute the diagonal elements \( M_{i,i} \) and \( K_{i,i} \)!

 
$$ M_{0,0}=L,\quad M_{i,i}=L/2,\ i>0,\quad K_{0,0}=0,\quad K_{i,i}=\frac{\pi^2 i^2}{2L},\ i>0\tp$$

 

Solving the equation system

 
$$ \begin{align*} Lc_0 &= Lc_{1,0} - \Delta t \cdot 0\cdot c_{1,0},\\ \frac{L}{2}c_i &= \frac{L}{2}c_{1,i} - \Delta t \frac{\pi^2 i^2}{2L} c_{1,i},\quad i>0\tp \end{align*} $$

 

 
$$ c_i = (1-\Delta t (\frac{\pi i}{L})^2) c_{1,i}\tp $$

 

We actually get a closed-form discrete solution:

 
$$ u^n_i = A(1-\Delta t (\frac{\pi}{L})^2)^n \cos(\pi x/L) + B(1-\Delta t (\frac{10\pi }{L})^2)^n \cos(10\pi x/L)\tp$$

 

Comparing P1 elements with the finite difference method; ideas

Comparing P1 elements with the finite difference method; results

Diffusion equation with finite elements is equivalent to

 
$$ \begin{equation*} [D_t^+(u + \frac{1}{6}h^2D_xD_x u) = \dfc D_xD_x u + f]^n_i \end{equation*} $$

 

Can lump the mass matrix by Trapezoidal integration and get the standard finite difference scheme

 
$$ \begin{equation*} [D_t^+u = \dfc D_xD_x u + f]^n_i \end{equation*} $$

 

Discretization in time by a Backward Euler scheme

Backward Euler scheme in time:

 
$$ [D_t^- u = \dfc\nabla^2 u + f(\x, t)]^n $$

 

 
$$ \begin{equation*} \uex^{n} - \Delta t \left( \dfc\nabla^2 \uex^n + f(\x, t_{n})\right) = \uex^{n-1} \end{equation*} $$

 

 
$$ \uex^n \approx u^n = \sum_{j=0}^{N} c_j^{n}\baspsi_j(\x),\quad \uex^{n+1} \approx u^{n+1} = \sum_{j=0}^{N} c_j^{n+1}\baspsi_j(\x)$$

 

The variational form of the time-discrete problem

 
$$ \begin{equation*} \int_{\Omega} \left( u^{n}v + \Delta t \dfc\nabla u^n\cdot\nabla v\right)\dx = \int_{\Omega} u^{n-1} v\dx + \Delta t\int_{\Omega}f^n v\dx,\quad\forall v\in V \end{equation*} $$

 

or

 
$$ \begin{equation*} (u,v) + \Delta t (\dfc\nabla u,\nabla v) = (u_1,v) + \Delta t (f^n,v) \end{equation*} $$

 

The linear system: insert \( u=\sum_j c_j\baspsi_i \) and \( u_1=\sum_j c_{1,j}\baspsi_i \),

 
$$ \begin{equation*} (M + \Delta t K)c = Mc_1 + \Delta t f \end{equation*} $$

 

Calculations with P1 elements in 1D

Can interpret the resulting equation system as

 
$$ \begin{equation*} [D_t^-(u + \frac{1}{6}h^2D_xD_x u) = \dfc D_xD_x u + f]^n_i \end{equation*} $$

 

Lumped mass matrix (by Trapezoidal integration) gives a standard finite difference method:

 
$$ \begin{equation*} [D_t^- u = \dfc D_xD_x u + f]^n_i \end{equation*} $$

 

Dirichlet boundary conditions

Dirichlet condition at \( x=0 \) and Neumann condition at \( x=L \):

 
$$ \begin{align*} u(\x,t) &= u_0(\x,t),\quad & \x\in\partial\Omega_D\\ -\dfc\frac{\partial}{\partial n} u(\x,t) &= g(\x,t),\quad & \x\in\partial{\Omega}_N \end{align*} $$

 

Forward Euler in time, Galerkin's method, and integration by parts:

 
$$ \begin{equation*} \int_\Omega u^{n+1}v\dx = \int_\Omega (u^n - \Delta t\dfc\nabla u^n\cdot\nabla v)\dx + \Delta t\int_\Omega fv \dx - \Delta t\int_{\partial\Omega_N} gv\ds,\quad \forall v\in V \end{equation*} $$

 

Requirement: \( v=0 \) on \( \partial\Omega_D \)

Boundary function

 
$$ u^n(\x) = u_0(\x,t_n) + \sum_{j\in\If}c_j^n\baspsi_j(\x)$$

 

 
$$ \begin{align*} \sum_{j\in\If} & \left(\int_\Omega \baspsi_i\baspsi_j\dx\right) c^{n+1}_j = \sum_{j\in\If} \left(\int_\Omega\left( \baspsi_i\baspsi_j - \Delta t\dfc\nabla \baspsi_i\cdot\nabla\baspsi_j\right)\dx\right) c_j^n - \\ &\quad \int_\Omega\left( \left(u_0(\x,t_{n+1}) - u_0(\x,t_n)\right) \baspsi_i + \Delta t\dfc\nabla u_0(\x,t_n)\cdot\nabla \baspsi_i\right)\dx \\ & \quad + \Delta t\int_\Omega f\baspsi_i\dx - \Delta t\int_{\partial\Omega_N} g\baspsi_i\ds, \quad i\in\If \end{align*} $$

 

Finite element basis functions

 
$$ \begin{align*} u^n &= \sum_{j\in\Ifb} U_j^n\basphi_j + \sum_{j\in\If}c_{1,j}\basphi_{\nu(j)},\\ u^{n+1} &= \sum_{j\in\Ifb} U_j^{n+1}\basphi_j + \sum_{j\in\If}c_{j}\basphi_{\nu(j)} \end{align*} $$

 

 
$$ \begin{align*} \sum_{j\in\If} & \left(\int_\Omega \basphi_i\basphi_j\dx\right) c_j = \sum_{j\in\If} \left(\int_\Omega\left( \basphi_i\basphi_j - \Delta t\dfc\nabla \basphi_i\cdot\nabla\basphi_j\right)\dx\right) c_{1,j} - \\ &\quad \sum_{j\in\Ifb}\int_\Omega\left( \basphi_i\basphi_j(U_j^{n+1} - U_j^n) + \Delta t\dfc\nabla \basphi_i\cdot\nabla \basphi_jU_j^n\right)\dx \\ &\quad + \Delta t\int_\Omega f\basphi_i\dx - \Delta t\int_{\partial\Omega_N} g\basphi_i\ds, \quad i\in\If \end{align*} $$

 

Modification of the linear system; the raw system

 
$$ \begin{align*} \sum_{j\in\If} \biggl(\underbrace{\int_\Omega \basphi_i\basphi_j\dx}_{M_{i,j}}\biggr) c_j &= \sum_{j\in\If} \biggl(\underbrace{\int_\Omega \basphi_i\basphi_j \dx}_{M_{i,j}} - \Delta t\underbrace{\int_\Omega \dfc\nabla \basphi_i\cdot\nabla\basphi_j\dx}_{K_{i,j}}\biggr) c_{1,j} \\ &\quad \underbrace{+\Delta t\int_\Omega f\basphi_i\dx - \Delta t\int_{\partial\Omega_N} g\basphi_i\ds}_{f_i},\quad i\in\If \end{align*} $$

 

Modification of the linear system; setting Dirichlet conditions

 
$$ \begin{equation*} Mc = b,\quad b = Mc_1 - \Delta t Kc_1 + \Delta t f \end{equation*} $$

 

For each \( k \) where a Dirichlet condition applies, \( u(\xno{k},t_{n+1})=U_k^{n+1} \),

Or apply the slightly more complicated modification which preserves symmetry of \( M \)

Modification of the linear system; Backward Euler example

Backward Euler discretization in time gives a more complicated coefficient matrix:

 
$$ \begin{equation*} Ac=b,\quad A = M + \Delta t K,\quad b = Mc_1 + \Delta t f \end{equation*} $$

 

Observe: \( A_{k,k} = M_{k,k} + \Delta t K_{k,k} = 1 + 0 \), so \( c_k = U_k^{n+1} \)

Analysis of the discrete equations

The diffusion equation \( u_t = \dfc u_{xx} \) allows a (Fourier) wave component

 
$$ \begin{equation*} u = \Aex^n e^{ikx},\quad \Aex = e^{-\dfc k^2\Delta t} \end{equation*} $$

 

Numerical schemes often allow the similar solution

 
$$ \begin{equation*} u^n_q = A^n e^{ikx} \end{equation*} $$

 

Handy formulas

 
$$ \begin{align*} [D_t^+ A^n e^{ikq\Delta x}]^n &= A^n e^{ikq\Delta x}\frac{A-1}{\Delta t},\\ [D_t^- A^n e^{ikq\Delta x}]^n &= A^n e^{ikq\Delta x}\frac{1-A^{-1}}{\Delta t},\\ [D_t A^n e^{ikq\Delta x}]^{n+\half} &= A^{n+\half} e^{ikq\Delta x}\frac{A^{\half}-A^{-\half}}{\Delta t} = A^ne^{ikq\Delta x}\frac{A-1}{\Delta t},\\ [D_xD_x A^ne^{ikq\Delta x}]_q &= -A^n \frac{4}{\Delta x^2}\sin^2\left(\frac{k\Delta x}{2}\right) \end{align*} $$

 

Amplification factor for the Forward Euler method; results

Introduce \( p=k\Delta x/2 \) and \( C=\dfc\Delta t/\Delta x^2 \):

 
$$ A = 1 - 4C\frac{\sin^2 p}{1 - \underbrace{\frac{2}{3}\sin^2 p}_{\hbox{from }M}}$$

 

(See notes for details)

Stability: \( |A|\leq 1 \):

 
$$ \begin{equation*} C\leq \frac{1}{6}\quad\Rightarrow\quad \Delta t\leq \frac{\Delta x^2}{6\dfc} \end{equation*} $$

 

Finite differences: \( C\leq {\half} \), so finite elements give a stricter stability criterion for this PDE!

Amplification factor for the Backward Euler method; results

Coarse meshes:

 
$$ A = \left( 1 + 4C\frac{\sin^2 p}{1 + \frac{2}{3}\sin^2 p}\right)^{-1} \hbox{ (unconditionally stable)} $$

 

Amplification factors for smaller time steps; Forward Euler

Amplification factors for smaller time steps; Backward Euler