I have read joint paper:
AlexSimmons, QianqianYang∗, TimothyMoroney, A finite volume method for
two-sided fractional diffusion equations on non-uniform meshes,Journal
of Computational Physics 335 (2017) 747–759,
and I try to implement the first test problems in this paper. I would
appreciate you if you answer my questions:
I need the initial value $u_0$ in recurrence relation (24), l use
the exact solution (46) (Test problem 1) at time zero for different
values of $x$ but l can not obtain the value of $u(x,0)$, then l try
to use the remark about the exact solution and Dirac $u(x, 0) =δ(x).$
and l put $u(x, 0) =0$ for nonzero $x$ and l do not know the value
of $u(0,0)$.
Kindly, guide me to choose the initial vector $u_0$ in (24).
I will appreciate you for guiding me to choose the initial vectors.
Kind Regrads,
Zeinab,
Things are easier when you use a (locally) conservative algorithm. This is especially the case when your IC are discontinuous. You can implement the delta function in at least three ways, a) you can simulate the delta function with a skinny triangle of pyramid (depending on the number of dimensions), b) you can replace the delta function with a skinny normal density function, and c) you can subtract the analytical solution due to the delta function in the vicinity of the IC.
The easiest thing to do to get things going is to replace the delta function with a slender pyramid or triangle (depending on the number of dimensions) and get a feeling for how your algorithm responds. The width of the little pyramid must be much smaller than other relevant scales in the vicinity of the IC. In many cases this is all you will need, and doing anything fancier with a delta function isn't warranted since diffusion problems "forget" the finer details of their IC.
You have two numerical tricks to implement that:
1- Do not start the simulation form t=0, start it from t=t_1>0 and write Dirac Delta as a limit of Gaussian distribution when variance goes to zero (see Wikipedia).
2- Do not start simulation from t=0 and start from t>t_0, only give one cell/volume/element in the middle of the domain, an initial value of a large number, but not so large you break your numerical algorithm.
Hope it halps,
Kaveh
How about contacting the authors of the paper directly, by e-mail? For instance, Timothy Moroney is an associated professor at the Queensland University of Technology, https://staff.qut.edu.au/staff/t.moroney.
By the way, the correct definition of the Dirac delta function differs from that given in one of the above answers.
Please be informed that the Dirac Delta is not a function. It is categorized in the Generalized Functions not functions. This is the reason that you cannot use the properties of a function for the Dirac Delta.
For instance, for a function, one can find a correspondence between the values of x and y, which is defined by y=f(x). In other words, for any x0 in its domain, one can find y0 for which we have y0=f(x0). But this is important to note that for a generalized function such as the Dirac Delta, it is not true. Hence, it is not true to write for example delta (1), delta (0) and something like this. For a Generalized Function such as the Dirac Delta, just the Sifting Property is valid, which is an integral relation with a test function.
For a clear and simple introduction to the topic,
mathworld.wolfram.com/DeltaFunction.html
seems to be helpful.
There, one may also find hints for implementing the "Dirac delta function", when
doing numerics.
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