On the numerical approximation of viscosity solutions for the differential-functional Cauchy problem

We consider the Cauchy problem for first order differential-functional equations. We present finite difference schemes to approximate viscosity solutions of this problem. The functional dependence in the equation is of the Hale type. It contains, as a particular case, equation with a retarded and deviated argument, and differential-integral equation. Numerical examples to illustrate the theory are presented.

Throughout the paper C(D) stands for the space of all continuous functions w : D → R with the supremum norm · D .
Let f :¯ × C(D) × R m → R and ϕ : 0 → R be continuous functions. We will consider the Cauchy problem for the first order differential-functional equation in the following form In the above Du denotes a gradient of u with respect to the space variable x. We write Du, D t u, u for the values at point (t, x) and u (t,x) for the Hale operator.
Although (1) is formulated in a rather abstract way it contains as a particular case a large group of differential-functional equations. The most important are: equations with a retarded and deviated argument, differential-integral equations, and of course equations without functional dependence on u. All these situations can be derived from (1), (2) by specializing the function f .
The next example shows how to transform differential-integral problem to (1), (2).
Of course, we can combine these two kinds of functional dependence and treat them using one model. We can also multiply functional dependence in (3) and (4) by putting u(μ 1 (3) and introducing K 1 , . . . , K N in (4).
In this paper we will investigate viscosity solutions of (1), (2). and Definition 2 A function u ∈ C(E) is a viscosity solution of (1), (2) if u is both a viscosity subsolution and supersolution of (1), (2).
We use the symbol SO L( f, ϕ) for the set of all viscosity solutions of (1), (2). This notion of solution was first introduced in [2,15] for the first order differential equations. The second order equations (not considered here) are widely presented in [1]. The Cauchy problem for differential-functional equations is investigated in [20].
There are numerous papers concerning difference schemes for the first order equations where nonfunctional dependence and classical solution are investigated. Here we concentrate on functional problems where generalized solutions are treated. Numerical approximation for generalized solution of first order equation was first investigated in [16] for weak solutions (in distributional sense), and in [11][12][13] for almost everywhere solutions. (with a restrictive assumption of convexity in the last variable). Difference methods are used in [18] to prove the existence of weak solutions for quasilinear equations with functional dependence and in [6] for generalized solutions with entropy uniqueness condition (see [11][12][13]). The method presented leads to existence results rather than to practical applications. In the study of viscosity solution the convexity assumption can be released. We also do not need the additional assumption on the solution (entropy condition). Moreover, the difference scheme applied in purely theoretical papers [3,17], although giving a slow convergence (a square root rate), can be useful in practical experiments. In this paper we extend results obtained in [3,17] into the case of equation with functional dependence on u. We based our reasoning on the estimations obtained for the nonfunctional case and on the a priori estimations for the functional case ( [19]). We also present some numerical experiments where functional dependence leads to many practical difficulties.
Numerical approximation for classical solutions of first order equation with functional dependence was investigated in [4], where explicit method is considered and in [5,10], where implicit schemes are treated. Convergence of the difference analog of the first order equation is investigated in [14] via difference inequalities.

Finite difference scheme
In this part we present finite difference method to approximate viscosity solution of (1), (2).
. Let x α = αk and t n = nh for n ∈ Z. Define Of course, for fixed n we have U n : Z m → R and for fixed α, U α : We write BC(X ) for the space of all continuous and bounded real functions on X ⊆ R k and BC(X ; L) for the space of all u ∈ BC(X ) Lipschitz with a fixed constant L > 0.
For X ⊆ R 1+m we define X t = {(s, z) ∈ R 1+m : s ≤ t} and write for short In view of the standard construction of T (see [9]) we can assume that T : Put γ, β ∈ Z m such that γ, β ≥ 0. Let p be a number of δ ∈ Z m satisfying inequalities α − γ ≤ δ ≤ α + β. For given g :¯ × C(D) × R mp → R and = ϕ | 0 we define a finite difference scheme: Of course, the above scheme always has a solution. We will write U ∈ AP(g, ) if U solves (10), (11) nevertheless AP(g, ) admits only one element.
Assume now that g is independent of w. We will write for short Scheme (10), (11) takes now the form In this paper we will consider only the monotone schemes, i.e. such that G(s, U ) is nondecreasing function of U (the exact definition will be given in the next section). This property of G is justified by the following, Proposition 1 Let G be defined by (12) and s = 0, 1, . . . , N . Then, Proof As (1) is immediate we begin with (2).
By changing the role of U and V we obtain the desired estimation. Now we will demonstrate (3). Let This and (2) imply (3) is proved.
Proof In view of Proposition 1 (2) we have By repeating this n-times we obtain the desired inequality.

Convergence of the scheme
In this section we will consider a general situation when f depends on w. Particularly interesting is the case when this dependence is functional. Our results can be applied to a large class of differential-integral equations and equations with a retarded argument (see Examples 1 and 2).

Assumption 1 (A) Suppose that,
(1) there exists γ > 0 such that | f (t, x, 0, 0)| ≤ γ in¯ , (2) f (t, x, u, p) is global Lipschitz continuous in u and local Lipschitz continuous in p, Here C L (D) stands for the space of all Lipschitz functions on D.
Since we use the space C L (D) in (3) and (4) we can apply our results to equations with a retarded and deviated argument under restriction that α depends on t and β depends on x. It would be impossible if we considered the space C(D) leaving out L x [u], L t [u]. Of course, the assumption would be stronger in this case, general enough to cover only differential-integral equation and constant retarded and deviated argument.
Assumption A can be formulated in more general form (see [19]) which gives a priori estimations on the solution and its Lipschitz constant in x (with a natural assumption on ϕ). Such general formulation can be reduced, however, to our formulation by a standard argument. Now we will investigate the finite difference scheme (10), (11).

Definition 3
We say that g is consistent with f if for every a ∈ R m g(t, x, u, a, . . . , a) = f (t, x, u, a) in¯ × C(D).
In the following we will assume that g is consistent with f , and k i / h for i = 1, . . . , m are constant. Put λ x i = k i / h. In view of [19] we know that if f satisfies Assumption A, ϕ ∈ BC( 0 ; L 0 ),ũ ∈ SO L( f, ϕ), then there exists L > 0 independent ofũ such thatũ ∈ BC(E; L). Let L be such a constant.
Let G[u] be defined by (12) with g replaced by g [u].

Definition 4
We say that scheme (10) The main result of our paper is

Numerical examples
Put m = 1. Let us consider the following Lax scheme: Here we have γ = 1, β = 0, p = 2, α = j. Since (18) can be written in the form we define where λ = k h . It is easily seen that g is consistent with f . Now we will describe what it means that our scheme is monotone on We  Table (a) below. The last two columns represent the case of non-monotonic scheme. From Table (b) we can see that errors (in monotonic case) satisfy a theoretical estimation given in Theorem 1. (K 1 = Ae Cτ ≤K ). In fact the theoretical error can be quite large for a large time interval. It is due to the fact that it grows exponentially in time and depends on a priori estimates. For instance an a priori Lipschitz constant L for the solution is much greater here than the real one.  An interesting effect can be observed if we prolong the time interval beyond π . The error estimate is growing up. It is due to the fact thatũ(t, x) = cos (t − 2|x|) is no longer a viscosity solution for t > π. (it is still a.e. solution). Our method gives an approximation of viscosity solution which exists and is unique globally. It is rather difficult to find an explicit formula for such solution if t > π. Maximal errors in the set [3,5]  Example 4 Consider the Lax scheme for the following problem where  Table (a) below. The last two columns represent the case of non-monotonic scheme. Numerical errors can be compared with the theoretical results by using where We verify thatũ such thatũ(t, x) = − arctan (|x| − t 2 ) in [0, 2] × R and satisfying (25) is a viscosity solution of the problem. The monotonicity condition for the scheme (18)  Example 6 Now we will give a numerical solution to the problem for which the exact solution is not known. Consider the Lax scheme for: The monotonicity condition for the scheme (18)  It is easily seen that violation of the monotonicity condition (the last column) leads to significant errors. It could also be seen from the graph of approximate solution. In Fig. 1 graphs of two solutions for which the monotonicity condition holds, are displayed. These graphs are very similar (we see both as almost one graph). Figure 2 represents the case for which the optimal link is not satisfied.
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