FINITE DIFFERENCE APPROXIMATIONS FOR A CLASS OF NON-LOCAL PARABOLIC EQUATIONS

ABSTRACT. In this paper we study finite difference procedures for a class of parabolic equations with non-local boundary condition. The semi-implicit and fully implicit backward Euler schemes are studied. It is proved that both schemes preserve the maximum principle and monotonicity of the solution of the original equation, and fully-implicit scheme also possesses strict monotonicity. It is also proved that finite difference solutions approach to zero as t --, oo exponentially. The numerical results of some examples are presented, which support our theoretical justifications.

Here $(z,t) is the temperature, v(x,t) is the displacement component in the direction of the z-axis, 0 is a uniform reference temperature, ct is the coefficient of expansion, and A, # are the elastic moduli.The boundary conditions are 0(-L, t) 0(L, t) Oo, (-L, t) (L, t) 0. For the detail derivations of the above equations we refer to [1, 3, 5].
In this article we study finite difference schemes for (1.1).The finite difference procedures proposed below preserve monotonicity, the maximum principle and the exponential decay (if the kernel is non-negative) of the solution for equation (1.1); therefore, they are considered as good numerical approximations.
Let h Az Ay 1/N for some integer N > 1, and let r > 0 be a small step-size in time with t. nr, n 0,1,....For a smooth function v(x, y) E C2() we assume that the following numerical integration formula is valid: N K(x, y, , ?)v(f, ?)dfd7 w,.,,K(z, y, xm, y) v,,, + O(h2), where w,,,, > 0 are weights and v,,, v(z,,,y) with x,, max, y IAy, rn, 0,1,...,N.For any 0 < p" < 1, we restrict h to be so small, say for some h0 > 0, 0 < h _< h0, that Here and throughout this paper we assume that h is small enough so that (1.10) is satisfied.In fact (1.10) serves as a discrete version of (1.2).In order to obtain the numerical solution which preserves as many properties of the solution as possible, (1.10) is a necessary and cannot be considered as a constraint on space discretization.For example the weights can be chosen by using trapezoidal rule, AxAy, w,,,t 1 / 4 AzAy, m,l 1,2,...,N-1; rn, 6 {0,g}; otherwise.
Clearly, (1.14) results in an error O(h +r) and is easy to implement numerically since only a pent- diagonal matrix system needs to be solved at each time level.Therefore, it is a very economical and fast algorithm.In addition, it is also unconditionally stable.Alternative methods, say ADI, may also be used to solve (1.14).
We observe that U" >_ 0 for all n >_ 0. Consider the first two levels n 0 and n 1. Assume to the contrary that U _< U1, then U > 0. If U IUio,j, Uo,j, > 0 the case Uo,j < 0 can be treated in a similar way) for some (i0,j0), then it follows from the discrete maximum principle [7] that this maximum is attained at the boundary.Thus (io,jo) can be selected to be a boundary point.Then, we see from (1.10), (1.12) and (1.13) that (2.4) U Ig,,jo ({U,,}) _< p'U which is impossible unless U 0 since p < 1.This contradicts U > 0. By (1.12), U 0 will lead to U . .0 for i,j 1 2,..-N 1 which is a contradiction Thus, 0 < U < U I, Now we consider the levels n 1 and n 2. By repeating the above argument with U . .s the initial data, we can show that 0 < U < gl.Thus, (2.8) is proved by repeating the above argument for higher levels.
The remainder of the proof follows from an argument similar to the above and mathematical induction. Q.E.D.
REMARK.Theorem 2.1 and Theorem 2.2 imply the unconditionally stability of numerical solutions U .n .,,jand W.. n.,,J, even though W. .n ., , is the solution of semi-implicit finite difference scheme.
In [8] b'Yiedman proved that V(t) decays exponentially when (1.2) is satisfied.We have proved that both {U'} and {W'} possess the strict monotonicity.In fact numerically there exists , > 0, as suggested in the examples of section 5, Figure 6 and Figure 7, such that Un+l (2.8) log U" ,,-AAt as n --, oo, and same is true for Wn.This motivates the justifications of the exponential decay of U and W when the kernel is non-negative.THEOREM 2.3 Under the assumption that U .n. is the solution of (1 12)-(1.13)and the t,.I kernel K(x, y, , r/) _> 0, there exists a positive constant A > 0 such that for U maxi,j [U 0.'JI, (2.9) U" _< U0e -xt" for all n _> 0.
PROOF.Let V(z, y, t) e-xt(2U (z 2 + V2)) where e and are two positive constants to be chosen below.It follows easily that there exists e0 > 0 such that e y U0 V(x,y,O)=2UO--(x + )> on f if 0<e<eo.
PROOF.The proof consists of an argument similar to that given in the proof of Theorem 2.3, we therefore only give the outline.
Let V(z, y, t) e-t(2W 2 y (z + )) and as before, let e0 be chosen so small that V(z, y, O) > W for 0 < e _< e0.Because the numerical integration uses the data on the previous level for the boundary condition, we need to first select a ,k0 > 0 such that 0 < A _< A0, e -r > p* i.e. e-'"2W > p*2W Since e -r --1 when , 0, the existence of such a )o is not a problem.With eo and Ao chosen as above, we then select el > 0 so small that for 0 < e _< el Using e min{e0, el } and the A0 selected above, we select A1 > such that for 0 < A < A, ,n > O, i,j=l,2,...,N-1.
We take A min{A0, A }. We omit the reminder of the proof which is the same as that given in the proof of Theorem 2.1 with the e and A as chosen here. Q.E.D.

CONVERGENCE AND ERROR ESTIMATES.
In this section we study the convergence and error estaimes of the numerical procedures proposed in section 1.First, we show the following result.

FINITE DIFFERENCE APPROXIMATIONS 155
We now show that there exists C > 0 such that Z.".< C(h 2 + r) for all i,,n.If Z" has a positive maximum, then according to the discrete' maximum principle then it must be attained at a boundary point.Assume that jr Z70i0 > 0 with no _> 1, is the positive maximum.From the boundary condition in (3.7) we see that M < p'M + (,o" + l)ma.x1o,l + m ,, p" + 3 Lo(h + r), <p'M+ 2 which implies p" + 3 Lo(h 2 + r).
In this section we consider the effect on the original problem (1.1) when the kernel condition (1.2) is replaced by: (4.1) 0 < K(x,V,,y) < Ko, (z,V,,y) E On x In general if the condition (1.2) is not satisfied, then the numerical procedure of (1.12) or (1.14) may not be stable uniformly for 0 < t < oo.This will be demonstrated in both theoretically and through numerical examples below.For these kernels, the stability will depend upon K0 and T > 0.Here we consider a class of kernels which satisfy (4.1) but not (1.2).
For w(x,y) chosen in this way, we have for some K K(Ko) > 0 that I,'X/l K1.Now consider the transformation v(x,y,t) =' eXtY(:r., y,t) with A _> KI, we find that Y satisfies (4.10) in QT, (x, ) e n (x,t)eOn, t>o.REMARK.We now see from [5] that Y possesses the maximum principle, monotonicity and exponential decay properties, which in turn results in monotonic and stable numerical schemes if it is discretized as (1.12) or (1.14) in an appropriate way.
Ikrning to numerical approximations for (1.1) with condition (4.1), we let " T/Nz where Nz is a positive integer.Numerical solutions to the problem, ,,j or ,,j, are defined as in (1.12)   or(1.14).We cannot expect that these two schemes have the monotonic properties as described in Theorem 2.1 and Theorem 2.2 when (1.2) is not satisfied.However, we have the following local stability estimates.THEOREM 4.1 Assume that U.". is defined as in (1.12) or (1.14) for the problem (1.1) with/t >_ 0 and (4.1) satisfied.If the solution u of (1.1) is known apriori to be smooth enough, u E C4'2(T), then there is some constant C* C*(]]ullc,.2, ]]/tl[c2,/to, T) > 0 such that PROOF The proof is similar to that given in section 3, so is outlined as below For (4.11), we let U.". _xt,.vn ,, wi,j,i,j, where A and w(x,y) are defined as above.Thus, it follows from a simple calculation that Y..".satisfies a difference equation which is the discrete version of the equation (4.10).Thus it follows from Theorem 3.1 and Theorem 3.2 (The proof needs only minor modifications from that given in Section 3, we therefore omit.) that there exists a positive constant C > 0 such that (4.12) max i,j Y(zi, t,frt)l < C'( h2 + i,j,n where C is indepedent of K0 and T > 0, and then, we obtain that (4.13) IU"'wid //, + r), ,,a u(zi,/j,t.)l< e x'" IU,-.(=i, t.)l < C*(h 2 which completes the proof.
Q.E.D. REMARK.The constant C" above can be very large if/to and T > 0 are very large.This can be seen from the choices of d and Kz in the above analysis, and also is demonstrated in the examphs in section 5.In another words although h and r are small, the error could be very big, even approaching oo as n --* oo.

NUMERICAL EXAMPLES.
We shall report several numerical examples which support our theoretical justifications in the previous sections, i.e., stability, monotonieity and exponential decay as --, co.Both semi- implicit and fully explicit schemes using trapezoidal rule for numerical integration are used in our computations.EXAMPLE 1.In order to demonstrate the error analysis and stability, we select f [0,2r] [0,2r], (z,Z/) sin(z)sin(y) and K(z,Z/,,7) -.Thus, for any real constant k > 0, u(z,Z/,t) sin(z)sin(z/)e -2t is the solution with fn IK(z,Z/,,r)ldrl k. Figure 1 and Figure 2 show by using semi-implicit scheme that the error distributions of u the maximum error on each level via the time) with the various parameter k from 0.1 to 4. Clearly, for k 0.1, 0.3, 0.5 and/ 0.8, even/ 1.0, the errors are under control as predicted by Theorem 3.1 On the other hand, for k 1.5, 2.5, 3 and k 4, it is seen that the errors are under control only for a short period of time, and then divergent to oo as n --, oo.This is the exact same result as predicted by Theorem 4.1, i.e., the numerical schemes are stable locally depednent upon K0 > 0 and T > 0. Figure 3 shows the error distribution of u by using fully implicit scheme.For 0 </ < 1 the error distributions of u in this example are almost identical to the case of/ 1.Also we noticed that the fully implicit scheme is more stable than the semi-implicit scheme.EXAMPLE 2. We now take a simple model problem with the same spatial domain and kernel as in example 1, $(z,V) ,in(zv) and/ 0.8. Figure 4 and Figure 5, by using semi- implicit and fully implicit schemes respectively, shows the distribution of U" via the time t, which decrease to zero exponentially as f -oo.If we assume roughly that for some A(), G() such that u(,) c(,). (')' .. ,oo, then A(f) can be calculated by the following formula Figure 6 and Figure 7, by using semi-implicit and fully implicit schemes respectively, show the distributions of A(t) proposed above, and it is seen that A approaches to a negative constant as expected.For semi-explicit scheme we find A n -0.145, and fully explicit A n -0.1336, thus the difference is 1.2 x 10 -2 which is within the rate of the truncation error of the discretization.
With A calculated above we then can compute C(t) by Figure 8 shows the distribution of C(t) computed by semi-implicit scheme according the above assumption.In this example we see that C(t) also approaches to a constant.Figure 9 and Figure 10 are the numerical solutions of u at t 0.5 and t 1.0 with h r/20 and r 0.01.EXAMPLE 3. Taking the same model problem as in example 2 except that the initial data $(z, V) (r z)(r V) and k 0.4.Figure 11, Figure 12 and Figure 13 show the distributions of U(t), A(t) and C(t) using the semi-implicit scheme.It is noticed that U(t) goes exponentially to zero very rapidly as t --, oo compared to that in example 2, this is due to that C(t) also approaches to zero, not a fixed constant as in example 2.
Prom these examples We have a rough idea how U(f) will behave as the time advances, i.e., we can at least by using numerical methods, semi-implicit or fully implicit scheme, to estimate the parameter A mentioned in Section 1.