# Convergence issue with time derivative discretisation **URL:** \ **Category:** variational formulation\ **Created:** [January 6, 2020, 9:20am UTC](https://fenicsproject.discourse.group/t/convergence-issue-with-time-derivative-discretisation/2167 "2020-01-06T09:20:31Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![RemDelaporteMathurin](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/remdelaportemathurin/32/4543_2.png) [@RemDelaporteMathurin](https://fenicsproject.discourse.group/u/RemDelaporteMathurin)\ **Post date:** [January 6, 2020, 9:20am UTC](https://fenicsproject.discourse.group/t/convergence-issue-with-time-derivative-discretisation/2167/1 "2020-01-06T09:20:31Z")
Hi all, I’d like to solve this system of PDE : ![image](https://global.discourse-cdn.com/free1/uploads/fenicsproject1/original/1X/dca5199161f92d43a990e7ddb5c5631741571d85.png) I ended up with this kind of formulation : ```python from fenics import * mesh = UnitIntervalMesh(10) V = VectorFunctionSpace(mesh, 'P', 1, 2) u = Function(V) u_n = Function(V) v = TestFunction(V) sol = list(split(u)) prev_sol = list(split(u_n)) test = list(split(v)) F = (sol[0] - prev_sol[0])/2*test[0]*dx - 1 * test[0]*dx F += (sol[0]*sol[1] - prev_sol[0]*prev_sol[1])/2*test[1]*dx - 1 * test[1]*dx solve(F == 0, u, []) ``` But this doesn’t converge (residual is nan). I reckon there’s something I’m not doing right here. Thanks for the help ! Rem ---
**Author:** ![kamensky](https://avatars.discourse-cdn.com/v4/letter/k/e95f7d/32.png) [@kamensky](https://fenicsproject.discourse.group/u/kamensky)\ **Post date:** [January 6, 2020, 3:37pm UTC](https://fenicsproject.discourse.group/t/convergence-issue-with-time-derivative-discretisation/2167/2 "2020-01-06T15:37:35Z")
Think about what the left-hand side matrix looks like in the first iteration. Since `sol[0]` is zero, the mass matrix that you get from linearizing the `sol[1]*test[1]` term is multiplied by a zero coefficient, and all the rows corresponding to `test[1]` are zero. ---
**Author:** ![RemDelaporteMathurin](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/remdelaportemathurin/32/4543_2.png) [@RemDelaporteMathurin](https://fenicsproject.discourse.group/u/RemDelaporteMathurin)\ **Post date:** [January 8, 2020, 8:42am UTC](https://fenicsproject.discourse.group/t/convergence-issue-with-time-derivative-discretisation/2167/3 "2020-01-08T08:42:58Z")
Ah that makes sense. So I suppose I should developp the time derivative term above as : ```python (sol[0]*(sol[1] - prev_sol[1]) + sol[1]*(sol[0] - prev_sol[0]))/2*test[1]*dx ``` is that right ? ---
**Author:** ![kamensky](https://avatars.discourse-cdn.com/v4/letter/k/e95f7d/32.png) [@kamensky](https://fenicsproject.discourse.group/u/kamensky)\ **Post date:** [January 8, 2020, 3:03pm UTC](https://fenicsproject.discourse.group/t/convergence-issue-with-time-derivative-discretisation/2167/4 "2020-01-08T15:03:07Z")
That still has the problem of the `sol[0]`–`test[1]` block in the LHS matrix being scaled by zero. There is now a `sol[0]`–`test[1]` block, but it is scaled by `sol[1]`, which is also zero, so this reformulation still crashes in the same way. Another option might be to have `soln[1]` represent C\_1C\_2, then recover C\_2 as a post-processing step (although that will become undefined when C\_1=0). This would completely decouple the system mentioned in the original post, but I assume it’s really an MWE for something more complicated, where the change-of-variables might not make sense. ---
**Author:** ![RemDelaporteMathurin](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/remdelaportemathurin/32/4543_2.png) [@RemDelaporteMathurin](https://fenicsproject.discourse.group/u/RemDelaporteMathurin)\ **Post date:** [January 8, 2020, 3:05pm UTC](https://fenicsproject.discourse.group/t/convergence-issue-with-time-derivative-discretisation/2167/5 "2020-01-08T15:05:46Z")
It seems that the following is working: ```python from fenics import * mesh = UnitIntervalMesh(10) V = VectorFunctionSpace(mesh, 'P', 1, 2) u = Function(V) u_n = Function(V) v = TestFunction(V) sol = list(split(u)) prev_sol = list(split(u_n)) test = list(split(v)) F = (sol[0] - prev_sol[0])/2*test[0]*dx - 1 * test[0]*dx # F += (sol[0]*sol[1] - prev_sol[0]*prev_sol[1])/2*test[1]*dx - 1 * test[1]*dx F += sol[0]*(sol[1] - prev_sol[1])/2*test[1]*dx - 1 * test[1]*dx + 1*sol[1]*test[1]*dx solve(F == 0, u, []) ``` what could be the issue then ? ---
**Author:** ![kamensky](https://avatars.discourse-cdn.com/v4/letter/k/e95f7d/32.png) [@kamensky](https://fenicsproject.discourse.group/u/kamensky)\ **Post date:** [January 8, 2020, 3:11pm UTC](https://fenicsproject.discourse.group/t/convergence-issue-with-time-derivative-discretisation/2167/6 "2020-01-08T15:11:23Z")
The `1*sol[1]*test[1]*dx` is giving you a nonzero mass matrix in the 11 block of the LHS matrix, although I’m not seeing how it follows from applying the product rule to the second equation. ---
**Author:** ![RemDelaporteMathurin](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/remdelaportemathurin/32/4543_2.png) [@RemDelaporteMathurin](https://fenicsproject.discourse.group/u/RemDelaporteMathurin)\ **Post date:** [January 8, 2020, 3:13pm UTC](https://fenicsproject.discourse.group/t/convergence-issue-with-time-derivative-discretisation/2167/7 "2020-01-08T15:13:11Z")
it is equal to the time derivative of sol[0] ---
**Author:** ![kamensky](https://avatars.discourse-cdn.com/v4/letter/k/e95f7d/32.png) [@kamensky](https://fenicsproject.discourse.group/u/kamensky)\ **Post date:** [January 8, 2020, 3:14pm UTC](https://fenicsproject.discourse.group/t/convergence-issue-with-time-derivative-discretisation/2167/8 "2020-01-08T15:14:31Z")
Okay, that makes sense.