Time dependent problems are commonplace in physics, chemistry and many other disciplines. In this post, I’ll introduce the FE formulation of linear time-dependent problems and derive formulas for explicit and implicit Euler integration.

The weak formulation of a first order time-dependent problem reads:

Find $u \in V$ such that

for all $v \in V$ and $t \in [0,\infty)$.

We can convert \eqref{eq:timedependentweak1} into a system of equations

where the components of the matrices and vectors involved are calculated as

If we further discretize in time with the finite difference $\dot{u} \approx [u_{n+1}-u_n]/{\Delta t}$, linearity allows us to write

This reflects on the system as

Here, $u_{n+1}:= u(x, t_{n+1})$, $\BM_{n+1} = \BM(t_{n+1})$ and vice versa for $u_n$ and $\BM_n$.

## Explicit Euler Scheme

For the explicit Euler scheme, we substitute evaluate the remaining terms at $t_n$

The corresponding system is

The update equation becomes

If $m$ is time-independent, that is $m(\dot{u}, v;t) = m(\dot{u}, v)$, we have

## Implicit Euler Scheme

For the implicit Euler scheme, we substitute evaluate the remaining terms at $t_{n+1}$

The corresponding system is

The update equation becomes

If $m$ is time-independent, one can just substitute $\BM=\BM_{n+1}=\BM_n$.

where $t\in I = [0,\infty)$,

• $\BD$ is a second-order tensor describing the diffusivity of $u$,
• $\Bc$ is a vector describing the velocity of advection,
• $r$ is a scalar describing the rate of reaction,
• and $f$ is a source term for $u$.

The weak formulation is then

Find $u \in V$ such that

for all $v \in V$ and $t \in I$.

We have the following integration by parts relationships:

for the diffusive part and

for the advective part. The canceled terms are due to divergence theorem and the fact that $v=0$ on the boundary. Then our variational formulation is of the form \eqref{eq:timedependentweak1} where

From these forms, we obtain the following system matrices and vector

where $\BM$ is constant through time.