Initial conditions for PDE

1. Apr 14, 2007

pavanakumar

I would really like to know whether initial conditions given to a time evolution PDE has to satisfy the governing equations. For example, if I have to solve numerically an incompressible flow equation do I need to give initial solution for the velocity field which is divergence free so as to obey the governing equation.

2. Apr 14, 2007

Chris Hillman

It sounds like you are asking about systems of PDEs which decompose into evolution equations plus constraint equations. If so, yes, the constraint equations place constraints on the initial values, and once you find a "legal" solution to those on some slice such as $t=0$, you use the evolution equations to obtain the complete solution. For example, you can consider Maxwell's source-free field equations to be a pair of evolution equations
$$E_t = \nabla \times \vec{B}, \; \; B_t = -\nabla \times \vec{E}$$
plus a pair of constraint equations
$$\nabla \cdot \vec{E} = 0, \; \; \nabla \cdot \vec{B} = 0$$
So initial data on the slice $t=0$ consists of two incompressible spatial vector fields $\vec{E}, \, \vec{B}$.

Last edited: Apr 14, 2007
3. Apr 18, 2007

pavanakumar

Thanks chris,
So my initial conditions must be mathematically consistent with the governing eqn.

I have one more query. If I want to give some sort of forcing in say E = f(t) (t -> time) along some boundary curve C. Then for each time I have no guarantee that the fields are divergence free. In this case too I must make the function f(t) divergence free. But how ? as I don't have prior knowledge of the fields at every time t (beforehand).