Initial value problems for PDE using finite difference method

I am new to numerical calculations. Recently I learned something about the finite difference method(FDM) and its applications in solving PDEs. For PDEs with initial and boundary conditions, I can understand the technique very well.

For initial value problems, we can obtain the solutions only according to the initial conditions. However, by means of numerical calculation, we cannot really deal with an infinite spatial domain. So we have to take a finite domain and add proper boundary value. My question is how to derive proper boundary values.

I can give a simple example. For advection equation $$\partial u/\partial t+\partial u/\partial x=0$$ with initial condition $$u(x,0)=f(x)$$.

Regardless of stability problems, we can use several FDMs. For example, $$u_{k,j+1}=u_{k,j}-\frac{\Delta t}{\Delta x}(u_{k,j}-u_{k-1,j}), \quad u_{k,0}=f_k$$. We cannot derive the values at all spatial points in the next time step according to the results of the previous time step.

I consulted some books on numeric PDEs, but I cannot find explanations for this problem.

• The concept of ghost point may be what you are looking for, they are a convenient way to impose BCs on all sides. Aug 28 at 12:49
• Randy LeVeque's book on finite difference methods for PDEs (staff.washington.edu/rjl/fdmbook) has extensive discussion on this topic using ghost points. I also use this book as a basis of class notes for my class on thee topic if that may be of use github.com/mandli/numerical-methods-pdes. Aug 29 at 14:15
• "we cannot really deal with an infinite spatial domain" FYI one trick is to change coordinates to a "compactified coordinate", which maps spatial infinity to a finite point Aug 29 at 16:54
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Aug 29 at 17:24
• @Kyle Mandli Thank you. I will consult the sources you recommended. I hope I can find an answer. Aug 30 at 8:51