[RT1995]

How to decide the initial or boundary conditions based on a givn partial diff equation or a set of PDEs.

Also,how to know where to choose them in a domain.

Also,how to choose whether derivative (Neumann) or actual value(Dirichlet) to use?

How to know how many to use?

[Moreza7]

Hello,

It depends on the physics of the problem.
For example in solving the heat equation you may have the constant temperature on a wall (Dirichlet B.C.) or the flux on the wall (Neumann B.C.) and the initial condition can be the equal to the.ambient temperature.

[RT1995]

Hey,

Thanks for your answer.

My specific question is as follows.

If a PDE is given or a set of PDE is given,can we decide how many IC and BCs to give from the order of the given individual terms? For example if there is 2nd order term then 2 BC.s ...etc.

[Moreza7]

Originally Posted by RT1995

Hey,

Thanks for your answer.

My specific question is as follows.

If a PDE is given or a set of PDE is given,can we decide how many IC and BCs to give from the order of the given individual terms? For example if there is 2nd order term then 2 BC.s ...etc.

As I know, the number of B.Cs for each variable is equal to the maximum order of its derivative in the differential equation which means a variable with a maximum derivative order of 2, needs 2 boundary conditions. Initial condition is one.


If anyone has more knowledge, please correct me.

[Simbelmynë]

Originally Posted by RT1995

Hey,

Thanks for your answer.

My specific question is as follows.

If a PDE is given or a set of PDE is given,can we decide how many IC and BCs to give from the order of the given individual terms? For example if there is 2nd order term then 2 BC.s ...etc.

If we assume that you are talking about CFD then the answer is not clear. It depends on the numerical scheme applied.


You need two boundary conditions for each spatial direction if you are using a central scheme since it demands information from both sides of a node (or face). If you use a forward or backward scheme then you only need one, since the numerical information only travels in one direction. A boundary condition that is located in the opposite direction from which the numerical information travels will never enter the calculation. So for pure convection, using upwind, you would only need one boundary condition for each spatial direction.


Diffusion is normally treated with central differences though, so if you have a convection-diffusion equation then you can assume that you need boundary conditions in each spatial direction. You also always need a numerical initial condition. For steady-state simulations, the choice of initial condition is mostly affecting the number of iterations you need to take to reach convergence. It does not have to have any physical meaning. If you have a transient simulation then your numerical initial condition also needs to be correct in the physical sense.

[LuckyTran]

We need to clear up some terminology. There isn't such a thing as a number of boundaries. There is a domain, of any shape, and it has extremities that we collectively call boundaries. The fact that there is a left boundary and right boundary in 1D problems doesn't make it two boundaries. Likewise in 2D shapes, if there are upper and lower boundaries doesn't make it 4 boundaries. There is simply a boundary.

The "number of boundary and initial conditions" needed to solve a particular problem is better to be referred to as the number of constraints needed to make the solution unique. How many you need can be determined from the general solution to the PDE you are solving. How to get the general solution? Set up the characteristic equation (i.e. solve the eigenvalue problem). The number of unknowns coefficients in the general solution is equal to the order of the characteristic polynomial. Yes the number of constraints needed is very strongly correlated with the order of the PDE, but formally it comes from the number of remaining unknowns in the general solution.


An ODE example:
the general solution to y''+y=0 is y(x)=Asin(x)+Bcos(x). Hence you need two constraints to determine the unknown coefficients A and B. PDE's also have this property, except that you consider a linear combination of linearly independent solutions that are constrained also by eigenvalues. A different type of ODE/PDE will have a different general solution and that tells you how many constraints you need. If your ODE is y'=y, the general solution is y(t)=Ce^t and you need one constraint to determine C.

The reason you usually need 1 constraint for temporal problems and 2 for spatial problems in most problems that you encounter is because the general solution for those is Ce^t and Asin(x)+Bcos(x)

By the way, an n-th order ODE can be converted into a system of n 1st order ode's. Similarly, an n-th order PDE can also be converted into a system of lower order PDE's.

Where or when these constraints are applied doesn't matter, they mathematically have a solution. They don't have to be initial conditions, they could be final conditions or intermediate conditions. Same with boundary conditions. You could apply internal constraints. But we are usually interested in problems that have these constraints as initial conditions or boundary conditions and so we talk about these more.

[Simbelmynë]

I would argue that you always need at least 6 boundary conditions for a Cartesian 3D problem, even if you use FVM (or FEM).


If you reduce the domain to the smallest component, a single computational cell, then you need conditions on all faces of the cell. A hexahedron celll is the most natural since it coincides with the derivation of the equations. With a tethrahedron cell you will have fewer faces, but any condition imposed on faces that are not normal to one of the Cartesian axes is composed of two or more components in the Cartesian axes. I would argue that this treatment is effectively imposing multiple boundary conditions on the face - one in each Cartesian direction.

If we ignore to explicitly impose a boundary condition at one or more faces, then it will be imposed implicitly as a zero-flux condition.

[RT1995]

Hey ,

Great answers here.

I wanted to ask that,

Lets say there is a system of 3 equations. And the unknown variables are u,v and P.
And these variables are a mixture of derivatives lets say from 1st to 3rd order. Just by looking at this equation system and finding the type of the equation(hyperbolic,parabolic and elliptic) can we decide the number and placement of initial and boundary conditions?

[LuckyTran]

First I want to say Simbelmynë has written some things that are not incorrect but is discussing a slightly different topic. Numerical schemes in CFD, are a discrete form of the continuous partial differential equation projected onto the computational grid: FVM, FD, FEM, etc. That's why, when you use different numerical schemes, you need different #'s of constraints from the neighboring cells. Here you are talking about each computational cell having it's own system of equations that need to be solved which is categorically different than the global system of PDE's that you want to solve over the entire domain. However, you're still talking about solving systems of equations. It's just a different context.

Originally Posted by RT1995

Hey ,

Great answers here.

I wanted to ask that,

Lets say there is a system of 3 equations. And the unknown variables are u,v and P.
And these variables are a mixture of derivatives lets say from 1st to 3rd order. Just by looking at this equation system and finding the type of the equation(hyperbolic,parabolic and elliptic) can we decide the number and placement of initial and boundary conditions?

Yes, you solve for the characteristic equation which is analogous to determining the general solution. The characteristic equation is the PDE version of the characteristic polynomial in linear matrix systems. Once you write down the system of equations (whether PDE's, ODE's, or algebraic systems), you (or your friend who is good at math) can go fetch the characteristic equation and that will tell you how many constraints you need. For linear PDE's with constant coefficients it is very straightforward to get the general solution.

The general solution belongs with the PDE itself. So if you've seen the general solution to a specific form of a PDE before at least once in your life, then you can indeed just look at that PDE and know right away how many IC's and BC's you need.

The constraint can be Dirichlet or Neumann or Robin or Cauchy or mixtures of them. You just need enough of them. Well all of these constraints are either on the function value or the 1st derivative.... Sometimes you encounter situations where you might want to apply constraints to the 2nd or higher order derivatives. Then how to decide if you need them or not? That's where the trick to convert a higher-order PDE into a system of lower order PDE's comes in handy. A constraint to a higher order derivative is then either a Dirichlet/Neumann constraint to the reduced order system. That's why the classification of BC types ends with Dirichlet/Neumann and we don't have fancy names for higher order ones.