Abstract
As mentioned in the previous chapter, modern finite difference schemes must (i) be at least of second order of approximation in all independent variables; (ii) be unconditionally stable; (iii) preserve nonnegativity of the solution. To achieve these goals, it became a common practice to involve a special apparatus of matrix theory that operates with so-called M-matrices. However, this approach is rather restrictive; e.g., we are unable to produce an M-matrix that is a second-order approximation of the first derivative. In this chapter, we show how some less well-known objects, namely Metzler matrices and EM (eventually M) matrices, can be used to eliminate these restrictions and provide a solid basis for building FD schemes with the declared properties. We give the main definitions and facts needed in the following to construct the appropriate finite difference schemes and prove the necessary theorems.
Morpheus: Do you know what I’m talking about
Neo: The Matrix.
Morpheus: Do you want to know what it is?
Neo: Yes.
Morpheus: The Matrix is everywhere. It is all around us. Even now, in this very room. You can see it when you look out your window or when you turn on your television. You can feel it when you go to work…when you go to church…when you pay your taxes…
The Matrix (1999)
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- 1.
This is especially important at the first few steps in time because of the step-function nature of the payoff. So a smoothing scheme, e.g., [18], is usually applied at the first steps, which, however, loses the second-order approximation at these steps.
- 2.
The trick is motivated by the desire to build an ADI scheme that consists of two one-dimensional steps, because for the 1D equations, we know how to make the right-hand-side matrix an EM-matrix [12].
- 3.
For the sake of clarity, we formulate this proposition for a uniform grid, but it should be fairly transparent how to extend it to a nonuniform grid.
- 4.
In our experiments, one to two iterations were sufficient to provide a relative tolerance of 10−6.
- 5.
Using the same β as above. However, changing the first multiplier on the right-hand side of Eq. (3.27) can make the scheme work for higher values of the time step as well.
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Itkin, A. (2017). An M-Matrix Theory and FD. In: Pricing Derivatives Under Lévy Models . Pseudo-Differential Operators, vol 12. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-6792-6_3
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