Información adicional: Atribuciones

For the sake of readability, we restrict to the one-dimensional equation (2.20), although the results of Theorem 8 apply to numerical schemes in any spatial dimension, as we will see in the next chapter, and we adopt the notation introduced in Section 2.1.

For the Chain Rule method described in Section 2.4.1, there is no way to impose the non-negativity of all stencil entries. Indeed, if we discretize the one-dimensional domain with points {xi}0≤i≤Nx, where ∆x = |xi+1−xi|

for a uniform mesh, and we consider the explicit Euler method for the time- discretization, also setting ai= a(xi) > 0 , ∀ i = 0, 1, ..., we have

un+1_{i − u}n i ∆t = ai+1− ai−1 2∆x · un i+1− uni−1 2∆x + ai un i+1− 2uni + uni−1 ∆x2 , (2.49)

so that the scheme in Table 2.1 reduces to un+1_i = ∆t ∆x2 a_i+1− a_i−1 4 + ai  un_i+1+_{1 − 2} ∆t ∆x2 ai  un_i + ∆t ∆x2  −ai+1− a₄ i−1 + ai  un_i−1.

In order to satisfy the non-negativity of the coefficients, we have to impose ∆t ∆x2 a_i+1− a_i−1 4 + ai  > 0 , ∆t ∆x2  −ai+1− a₄ i−1 + ai  > 0 , (2.50) independently from the choice of the diffusion function a(x), because we can always assume 1 − 2∆x∆t2ai > 0 by modifying appropriately the ratio between

time-step and space-step to fulfill the so-called parabolic CFL-condition ∆t < ∆x 2 2 amax , amax:= max 0≤i≤Nx ai. (2.51)

For the second condition in (2.50), we would have ai−

ai+1− ai−1

4 > 0 , ∀ i = 0, 1, . . . , Nx. We deduce from a Taylor’s expansion that ai+1−ai−1

4 = ∆x2 a′i+O(∆x3), with

abuse of notation if we denote by a′

i the value of the derivative at point xi.

Finally, the diffusion coefficients should satisfy the following constraint, ai> ∆x

2 a

′

i, ∀ i = 0, 1, . . . , Nx,

which alternatively gives an extra-restriction on the space-step ∆x (that is practically viable only for linear problems, i.e. in case of diffusion functions solely depending on space, and eventually time). We remark that a similar

constraint is deduced from the first condition in (2.50) and this concerns all possible signs for the derivatives. Therefore, if we approximate the parabolic operator on a coarse grid (as motivated by computational cost) and the dif- fusion function a attains low values over the domain but with rapid growth, i.e. ai is very small but its derivative is very high, the Chain Rule method

fails to be nonnegative, and the maximum principle can be violated. Remark 8. For the one-dimensional parabolic equation in non-conservative form with constant coefficients, i.e. ut = b ux+ a uxx, b ∈ R, a ∈ R+, we

can consider the finite difference centered scheme un+1_{i − u}n_i ∆t = b un i+1− uni−1 2∆x + a un i+1− 2uni + uni−1 ∆x2 ,

which is structurally similar to (2.49) since based on the same principle of separating the terms of different order, and we can rewrite it as

un+1_i = ∆t ∆x b 2 + a ∆x  un_i+1+_{1 − 2} ∆t ∆x2 a  un_i + ∆t ∆x  −b₂+ a ∆x  un_i−1.

Under the parabolic CF L-condition ∆t < ∆x_{2 a}2 , also corresponding to (2.51), the central coefficient above is positive and less than 1, moreover the sum of the coefficients is equal to 1 and they can be made all positive (independently from the sign of b) providing that the space-step ∆x is sufficiently small. Nevertheless, there is no reason why those coefficients should be less than 1 , thus preventing an L∞-stability of the scheme. Indeed, that scheme turns out to be a weakly-parabolic correction of the centered scheme for hyperbolic problems, which is well-known to exhibit oscillations [22], because the maxi- mum principle is not fulfilled. The L2-stability is guaranteed, however, as we can easily check by performing an analysis of its modified equation (obtained by Taylor’s expansions and manipulating the exact equation)

ut− b ux− a uxx− a 12∆x 2₋ a2 2 ∆t
uxxxx +b 2 2∆t ux+ a b ∆t − b 6∆x 2
_u xxx+ O(∆t2, ∆x5) = 0 ,

under some quite restrictive CF L-type condition, namely ∆t < ∆x_{6 a}2, which is the same predicted for the fully-parabolic problem in Section 3.1.4.

The limits of the Chain Rule method become insurmountable when pass- ing to two-dimensional problems : unless one deals with diagonal matrices, for which the issue discussed above is however relevant, the off-diagonal elements of the stencil in Table 2.1 cannot always be positive for fully anisotropic diffusion tensors, as the coefficients c may have any sign.

The discussion becomes more interesting for the Standard Discretiza- tionintroduced in Section 2.4.2, which equals the finite volume scheme (2.28) in the one-dimensional setting, as given by (2.31), thus resulting in a positive approximation. For two-dimensional problems, we can see in Table 2.2 that horizontal and vertical off-diagonal elements of the stencil are nonnegative, since a and b must be nonnegative from (1.7), but the non-negativity of the whole stencil cannot be guaranteed since c has undefined sign.

Nevertheless, the scheme in Table 2.2 enjoys certain stability properties, as reported in [7], because the approximate solution at any arbitrary fixed time remains bounded for some suitable norm defined on Ω . Typically, we con- sider the maximum norm and we say that the scheme is stable in (Ω, || · ||∞)

if there exists a constant CT > 0 such that ||un||∞ ≤ CT||u0||∞, ∀ n > 0 .

Explicit methods are often conditionally stable, i.e. if the time-step ∆t is chosen under certain CFL-conditions, as we will discuss in the next chapter. The Nonnegative method derived in Section 2.4.3 reduces to the stan- dard discretization, i.e. the finite difference scheme on staggered grids, in the one-dimensional case : this comes from the fact that the only modifica- tion needed to achieve non-negativity concerns the mixed derivatives, whose effect cannot be appreciated when no dimensional interaction is present. Then, it seems that the failure of the non-negativity property can actually occur whenever the scheme involves terms with mixed derivatives, and a nonnegative correction has to be introduced to control the coefficients at the vertices of Table 2.2, which are the only ones produced by the mixed derivatives. We will see in the next chapter that the nonnegative method in Table 2.4 satisfies all the properties listed in Table (2.5).

Chapter 3

Stability Analysis of

one-dimensional methods

In this chapter, we revisit details of classical methods in the one-dimensional setting, focusing on several finite difference/volume schemes, essentially in order to illustrate the main tools and strategies we will later apply to the study of two-dimensional problems.

We remark that most schemes introduced in the previous chapter coincide for one-dimensional equations, because the finite difference method on stag- gered grids can be reinterpreted as the finite volume approach, and for those schemes we check the validity of the properties listed in Table 2.5.