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I am reading the book 1 (Par 2.2 pag. 31) about the computational fluid dynamics. Unfortunately I don't understand the scheme of the characteristics of the wave equations. This scheme starts from the example of hyperbolic PDE:

$\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}=0$ (1)

The settled intial condition is:

$u(x,0)=sin(\pi x)$ , $\frac{\partial u}{\partial t }(x,0)=0$ (2)

The boundary condition is:

$u(0,t)=u(1,t)=0$ (3)

Here are my questions:

  1. How can I calculate the characteristic directions from (1)? Why the characteristics direction is $\pm\frac{d x}{dt }$?

  2. As far as I understood, the characteristic directions are the angular coefficients of the lines. why do they intercept the x-axis in $x_{i}-t_{i}$ and $x_{i}+t_{i}$? or maybe I am misunderstanding the Figure.

  3. Why the lower triangle is called "Domain of dependence" and the upper "Domain of Influence"?

I have tried to search other resources, but it is difficult to understand for me.

1 Fletcher, Clive A. J. (1998). Computational Techniques for Fluid Dynamics 1 . , 10.1007/978-3-642-58229-5()

user49811
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  • Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Oct 27 '22 at 11:31

1 Answers1

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(1) Different fields have different strategies for solving hyperbolic PDEs. You can check that the equation is solved by the sum of one wave traveling left at constant speed and another traveling right at constant speed.

(2) Those are the characteristics for just a single point.

(3) The domain of dependence indicates the region of the initial condition that impacts the solution at (x_1,t_1). It turns out with hyperbolic PDEs the propagation speed of information is finite. This matches the intuition that any events on the opposite side of the world have literally no impact on me until about .01 seconds later. Hence "domain of dependence". Domain of influence reflects the opposite phenomenon. Starting at (x_1,t_1), what regions of space/time are influenced by the function value, which once again reflects the idea that propagation is finite. I am only included in the domain of influence on events on the other side of the world about .01 seconds after they occur.

Here's a good video I'd recommend, short and to the point.

https://www.youtube.com/watch?v=Av1VJ5fJcKo

SimpleRick
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