The present article considers the numerical solution of ODE systems
stemming from the spatial discretization by using finite differences (or finite volumes) of semilinear parabolic PDEs with constant diffusion coefficients and an initial condition and Dirichlet Boundary Conditions (BCs) of the form
Here, is a source term and we assume that its discretization is entirely included in , so that , consist only of contributions from the boundary conditions in the
-direction. In particular, we shall be concerned with time-independent Dirichlet boundary conditions, in which case the vectorsare constant.
To prove convergence in the maximum norm for many numerical methods of splitting type applied to (1), it is customary to get uniform bounds for , where is the time stepsize and is a rational mapping acting on the matrices , . Typically, we have , with when second order central differences are considered in the spatial discretization of (2). Here, we denote the spacing , where is the number of equidistant grid-points on the -direction, and stands for the Kronecker product of matrices. It should be observed that the matrices pairwise commute. Such methods of splitting type when applied to (1) typically produce a recursion for the global errors , , of the form , where stands for the numerical solution at , denotes the local error and is the stability matrix associated to the numerical integrator. For ADI-type integrators the stability matrix depends on (see, e.g., (Hun-Ver03, , Sec. II.2.3)). A relevant example is
where , which has the associated stability function of complex variables
For the choice , this is the stability matrix of the Peaceman-Rachford method (when ), also the one of the Douglas scheme (Douglas62 , Hun-98a , (Hun-Ver03, , p. 373)) and the one of the one-stage AMF-W-method GHH-sinum20 . Furthermore, the stability matrix of the so-called Hundsdorfer–Verwer scheme (Hun-Ver03, , Section IV.5.2), which is a -stage W-method of order in general, and of order for , is given by
In this case the stability function is given by (with and defined in (4)). The power boundedness in the maximum norm of some -stability functions was already considered in GHH-BIT21 . However, for the power bound there obtained is not uniform, since it allows a logarithmic growth in terms of or GHH-BIT21 , i.e.,
With this power bound, convergence results in the maximum norm of size , when the local errors are of size can be obtained. However, with power bounds of the stability matrix as the one in (5), it can be shown convergence of size in case of time independent BCs in (2). In Section 2, we prove a result related to the power boundedness for rational functions. This result is applied in Section 3 to show unconditional convergence in the maximum norm for some ADI-methods. In Section 4, numerical experiments are included to illustrate the orders of convergence regarding the PDE solution for some relevant ADI-type methods.
2 Bounds in the maximum norm for rational functions
We look for bounds in the maximum norm of the form
where is a rational function (or a mapping when acting on the matrices ) of complex variables that is -stable, i.e.
Of course, if (6) holds true for some then it also holds for .
The proof of this theorem is given below and makes use of the following two lemmas.
For any matrix (5) it holds that
Proof. The formula in (7) is well known in the literature (see, e.g., (Mattheij, , formula (4.10)) or (lar-thom, , p. 43-45)). The formula (8) is an immediate consequence of Lemma 4.1 in GHH-BIT21 (see also (farago02ste, , formula (5))), with and , since
To show (9), by considering , we have that
To show (10), for it holds
Assume that for any positive integer we have that
Proof. The last two inequalities in (12
) follow from the fact that for positive numbers the Quadratic Mean is greater or equal than the Arithmetic Mean and this is greater or equal than the Geometric Mean. To show the first inequality, we observe that for complex numberssatisfying (11) it holds that Hence, fulfils and it has an angle with the negative axis. In particular, takes the form (11). Then, adding a new complex number (11) and using the same argument we deduce that fulfils and it has an angle with the negative axis. The application of the induction principle concludes the proof.
Proof of Theorem 1. We define and use below the following notation for the Kronecker product of matrices Consider the positively oriented boundary of the open domain , which is symmetric with respect to the negative real axis in the complex plane,
Observe that Let us define the rational function (and the associated mapping when acting on matrices)
Taking into account that is analytic if , from the Cauchy’s integral formula applied on each variable we get the following formula by using iterated integrals
By considering the mapping acting on the matrices we deduce that
Observe that the eigenvalues of each matrixare , , where
Hence, the spectrum of falls in .
At this point we should notice the identity From here, taking the maximum norm and using that for two matrices and , we get that
Next we bound when . We distinguish three cases.
, then , and . From (9) it follows that
and we deduce that
, then , and . From (8) it follows that
, then , and . From (10) it follows that
and we get that
From the A()-stability of we deduce that
Taking account that all these iterated integrals can be transformed into products of integrals in one variable, we get
Then, we have for ,
Hence, each term is bounded since
This concludes the proof.
3 Convergence in the uniform norm of some ADI-type methods
The first goal of this section is to show unconditional convergence of order two in the maximum norm for semilinear parabolic problems with constant diffusion coefficients (and a time dependent source term) and time-independent Dirichlet boundary conditions (1)-(2), when the one-step AMF-W-method (henceforth denoted as AMF-W1) in GHH-sinum20 ; gonzalez18amf is considered with the parameter choice
where stands for the derivative of a function regarding . The following discussion can be applied in similar terms to the Douglas method (Hun-Ver03, , p. 373). We use the same notations as in GHH-sinum20 . The global error at the time step is denoted as in (GHH-sinum20, , formula (2.3)) by where is the solution of the numerical method and is at the same time the exact solution of the (1) and the exact solution of the PDE on the set of discrete points of the spatial mesh-grid . Observe that we will not consider in our analysis the truncation errors introduced in the spatial discretization of the PDE, since when using central differences we get a stable space discretization and the truncated spatial errors do not play any important role in the analysis of global errors (space truncation errors plus time integration errors) as it can be seen e.g. in (Hun-Ver03, , Chapt. IV). It should be remarked that the discretization of the source term is entirely included in (GHH-sinum20, , Sect. 1). Besides, the terms
and they are smooth (i.e. they have bounded first and second derivatives independently of the spatial resolution), since is a smooth function and we have (below for and )
with , and the convention (see (GHH-sinum20, , formula (2.4))). We also make use of other expression for the global errors (see (GHH-sinum20, , formula (4.11))), obtained by partial summation in (28),
and of a simplified expression for the local errors given in (GHH-sinum20, , formula (4.7))
Since, from (26),