Hermite spectral and pseudospectral methods for nonlinear partial differential equation in multiple dimensions

Hermite spectral and pseudospectral methods for nonlinear partial differential equation in multiple dimensions

Xu Cheng-Long^I; Guo Ben-Yu^II

^IDepartment of Applied Mathematics, Tongji University, Shanghai, 200092, P.R. China

^IIDepartment of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. China

E-mail: [e-mail] / [e-mail]

ABSTRACT

Hermite approximation in multiple dimensions is investigated. As an example, a spectral scheme and a pseudospectral scheme for the Logistic equation are constructed, respectively. The stability and the convergence of the proposed schemes are proved. Numerical results show the high accuracy of this new approach.

Mathematical subject classification: 35A40, 65M12, 65M70 .

Key words: Hermite approximation, nonlinear partial differential equations, multiple dimensions.

1 Introduction

Spectral methods for partial differential equations in unbounded domains have been received more and more attentions recently. Gottlieb and Orszag [1], Maday, Pernaud-Thomas and Vandeven [2], Coulaud, Funaro and Kavian [3], Funaro [4], and Guo and Shen [5] developed the Laguerre spectral method. While Funaro and Kavian [6] provided some numerical algorithms by using Hermite functions. Furthermore, Guo [7] established some approximation results on the Hermite polynomial approximation with applications to partial differential equations. Guo and Xu [8] studied the Hermite pseudospectral method and obtained good numerical results.

As we know, most practical problems are set in multiple dimensions. We may set up some artificial boundaries and impose certain artificial boundary conditions, and then use the usual numerical methods to resolve them in bounded subdomains. But it is not easy to derive the exact boundary conditions, and so some additional errors occur usually. In opposite, if we use the Hermite approximation directly in unbounded domain, then the above trouble could be removed. However, so far, there is no results on the Hermite polynomial and interpolation approximations in multiple dimensions. The aim of this paper is to develop the Hermite spectral and pseudospectral approximations to nonlinear partial differential equations in multiple dimensions.

This paper is organized as follows. In Section 2, we establish some results on the Hermite polynomial approximation and Hermite interpolation in multiple dimensions which play important roles in the analysis of the Hermite spectral and pseudospectral methods. As an example, we construct a Hermite spectral scheme for the multiple dimensions Logistic equation and prove the stability and the convergence of the proposed scheme in Section 3. The corresponding pseudospectral scheme is discussed in Section 4. In the final section, we present some numerical results which show the high accuracy of this new approach.

2 Hermite approximation in multiple dimensions

In this section, we consider the Hermite approximation in multiple dimensions. Let L_i = {x_i|– ¥ < x_i < ¥}, L = L₁ × L₂ × ... × L_n, x = (x₁,x₂,...,x_n), |x| = , and w(x) = . For 1 < p < ¥, let

where

In particular, (L) is a Hilbert space with the inner product

Let k = (k₁,k₂,...k_n), |k| = k_i, k_i being any non-negative integers, and

For any non-negative integer m,

For any real r > 0, the space (L) with the semi-norm |v|_r,w and the norm ||v|| _r,w, is defined by space interpolation as in Adams [9].

Let l = (l₁,...l_n), l_i being any non-negative integers, and |l| = l_i. The Hermite polynomial of degree l is of the form

The set of Hermite polynomials is the (L)-orthogonal system, i.e.,

where l! = l_i! and

For any v Î (L),

where

Let N be any positive integer and _N be the set of all algebraic polynomials of degree at most N in each variable x_i,1 < i < n. The (L)-orthogonal projection P_N: (L) ® _N is a mapping such that for any v Î (L),

Let w_i(x_i) = and P_N,i be the (L_i)-orthogonal projection.

Lemma 2.1 (see Theorem 2.1 of Guo [7]). For any v Î (L_i) and 0 < m < r ,

We now consider the multiple-dimensional Hermite polynomial approximation.

Theorem 2.1. For any v Î (L) and 0 < m < r,

Proof. By (2.3) of Guo and Xu [8], P_N,i¶_xjv = ¶_xjP_N,iv for 1 < i, j < N. Therefore by Lemma 2.1,

In practice, we also need the (L)-orthogonal projection : (L) ® _N . It is a mapping such that for any v Î (L),

As explained in Guo [7], we can prove that the projection is exactly the same as P_N.

Next let

Lemma 2.2. For any v Î (L),

Proof. We have from integration by parts that for any i,

whence

Next,

By induction,

Similarly

Lemma 2.3. For any v Î (L),

Proof. We have

By induction,

where c_i are certain constants. Furthermore let y = (y₁,...,y_n). Then

By virtue of (2.1), Lemma 2.2 and the Cauchy inequality,

Theorem 2.2. For any v Î (L) and r > n,

Proof. By Lemma 2.3 and Theorem 2.1, we verify that

This completes the proof.

We now turn to the Hermite-Gauss interpolation. Let j = (j₁,...,j_n), 0 < j_i < N, 1 < i < n, and be the zeros of the Hermite polynomial H_N+1(x_i). Let

and L_N be the set of all points s_j. For any v Î C(L), the Hermite-Gauss interpolant I_Nv Î _N is determined by

Next let w^(j) be the Christoffel number with respect to w(x), namely,

where are the Christoffel numbers with respect to w_i(x_i), 1 < i < n.

We introduce the following discrete inner product and norm,

Clearly

For technical reasons, let

By Guo and Xu [8], if fy is a polynomial on L_i of degree at most 2N + 1, then

Guo and Xu [8] also proved that for any v Î (L_i),

By using (2.3), it can be shown that for any fy Î

_2N+1,

In particular,

Lemma 2.4. For any v Î (L),

Proof. We use induction. When n = 1, the desired result is exactly the same as (2.4). Suppose that the result is valid for n = m. Now let n = m + 1, and w_m(x) = . By virtue of (2.4), we have that

So

The induction is compete.

Theorem 2.3. For any v Î (L), 0 < m < r and r > n,

Proof. By Guo [10], for any f Î _N and m > 0,

We have from (2.5), (2.6) and Lemma 2.4 that

Therefore by Theorem 2.1,

Theorem 2.4. For any v Î (L) and r > n,

Proof. Thanks to Lemma 2.3 and Theorem 2.3, we get that for any x Î L,

The desired result follows.

We have from (2.5) and Theorem 2.3 that for any v Î (L),f Î _N and r > n,

3 Hermite spectral scheme for the logistic model

This section is for application of the Hermite spectral approximation to the Logistic equation in two-dimensions. We construct a Hermite spectral scheme, and prove its stability and convergence. The main idea and techniques in this section are also applicable to other nonlinear partial differential equations in n-dimensions.

Let y = (y₁,y₂) and L = (L₁, L₂). V(y,s) describes the population of budworm. g(y,s) and V₀(y) are the source term and the initial state of population, respectively. Then the Logistic model takes the form

As pointed out in [7], the Laplacian in (3.1) does not correspond to a positive-definite bilinear form in (L), and so (3.1) is not well-posed in the weighted space. So we take the following similarity transformation

Let

Then

So (3.1) becomes

Further, let

Then problem (3.3) is changed into

The weak formulation of (3.4) is to find U Î L²(0,ln(1 + T);(L)) Ç L^¥(0,ln(1 + T);(L)) such that

The Hermite spectral scheme for (3.5) is to find u_N(t) Î _N for all 0 < t < ln(1 + T), such that

We give some Lemmas which will be used in the analysis of the stability and the convergence of scheme (3.6).

Lemma 3.1 (see Lemma 2.3 of Guo [7]). For any v Î (L_i),

Lemma 3.2. For any v Î (L_i),

Proof. By integration by parts and Lemma 3.1, we obtain that

Lemma 3.3. For any v Î (L),

Proof. For any x Î L,

Similarly

Thus we have

The above with Lemma 3.2 leads to

We now consider the stability of (3.6). Assume that f and u_N,0 have the errors and _N,0, respectively. They induce the error of numerical solution u_N, denoted by _N. Then the errors fulfill the following equation

By taking f = 2

_N in (3.7), we obtain that

By the Schwartz inequality and Lemma 3.3,

Substituting (3.9) and (3.10) into (3.8) and integrating the result with respect to t, we obtain that

where

Lemma 3.4 (see Lemma 3.1 of Guo [7]). Assume that

(i) the constants b₁ > 0, b₂ > 0, b₃ > 0 and d > 0,

(ii) Z(t) and A(t) are non-negative functions of t,

(iii) d <

for certain t₁ > 0,

(iv) for all t < t₁,

Then for all t < t ₁ ,

Applying Lemma 3.4 to (3.11), we obtain the following result.

Theorem 3.1. Let 0 < a < 1 and u_N(t) be the solution of (3.6). If for certain t₁ > 0,

then for all t < t ₁ ,

Remark 3.1. Theorem 3.1 indicates that the scheme (3.6) is of generalized stability in the sense of Guo [11, 12], and of restricted stability in the sense of Stetter[13]. It means that the computation is stable for small errors of data.

We next deal with the convergence of scheme (3.6). Let U be the solution of (3.5), and U_N = P_NU = U. We get from (3.5) that

where

Let

_N(t) = u_N(t) – U_N(t). By subtracting (3.12) from (3.6), we get that

Comparing (3.13) to (3.7), we only need to estimate the terms |G_i(t,_N(t))|. By Theorem 2.1,

By Lemma 3.3 and Theorem 2.1, we have that for r > 1,

Finally we obtain the following result.

Theorem 3.2. If U Î H¹(0,ln(1 + T); (L)) with r > 1. Then for all 0 < t < ln(1 + T),

where c^* is a positive constant depending only on T and the norms of U in the spaces mentioned above.

Remark 3.2. By Theorem 3.2 and Theorem 2.1, we have that under the conditions of Theorem 3.2,

Remark 3.3. Since c₂(U_N, T) depends on T² linearly, we can see that c^* depends on T³ linearly.

4 Hermite pseudospectral scheme for logistic model

In this section, we consider a Hermite pseudospectral scheme for (3.5). Let n = 2, we use the same notations as in Section 2.

The Hermite pseudospectral scheme for (3.5) is to find u_N(t) Î _N for all 0 < t < ln(1 + T), such that

Remark 4.1. By (2.3), the first formula of (4.1) is equivalent to

The following Lemma will be used in the analysis of the stability and the convergence of scheme (4.1).

Lemma 4.1. For any v Î _N,

Proof. We have

Similarly

Therefore, by the Hölder inequality, (2.4) and Lemma 3.2, we obtain that

We now analyze the stability of (4.1). Assume that f and u_N,0 have the errors and _N,0, respectively, which induce the error of u_N, denoted by _N. Then the errors fulfill the following equation

Comparing (4.2) with (3.7), we only have to estimate the upper-bounds of the following terms with f =

_N,

By the Schwartz inequality, (2.5) and Lemma 4.1, we have that

Let

Then the following result follows.

Theorem 4.1. Let 0 <a < 1 and u_N be the solution of (4.1). If for certain t₁ > 0,

then for all t < t ₁ ,

where c₁(T) is the same as in Theorem 3.1.

Next, we deal with the convergence of scheme (4.1). Let U_N = P_NU = U. We get from (3.5) that

where

Further, let

_N(t) = u_N(t) – U_N(t). Then by (4.1) and (4.3),

Comparing (4.4) to (4.2), we only need to estimate |G_i(t,_N(t))|. Firstly, by Theorem 2.1,

Next, let

where

By Lemma 2.3 and Theorem 2.1,

Thus by (2.5), we have that for r > 2,

Due to (2.7),

It is easy to see that

where p_r(x_i) is a polynomial of degree at most r. By Lemma 2.3,

By Lemma 3.3 and the Schwartz inequality,

Hence

The previous estimates lead to

where c(T) is a positive constant depending only on T² . In addition, Theorem 2.3 implies that for r > 2,

Using Theorem 2.1 and Theorem 2.3,

Finally the following result follows.

Theorem 4.2. If U Î L⁴ (0,ln(1 + T); (L)) Ç H¹ (0,ln(1 + T); (L)), f Î L² (0,ln(1 + T); (L)) and U₀Î (L) with r > , then

where d^* is a positive constant depending only on T and the norms of U in the spaces mentioned above.

Remark 4.1. By Theorem 4.2 and Theorem 2.1, we have that under the conditions of Theorem 4.2,

Remark 4.2. Since c₂(U_N, T) depends on T² linearly, we can see that c^* depends on T³ linearly.

5 Numerical results

We present some numerical results in this section. We shall use schemes (3.6) and (4.1) to solve (3.5), respectively. The test function is

with a₁ = 0.3, a₂ = 0.3, a₃ = –0.1, a₄ = 3.0. In actual computation, we use the standard fourth order Runge-Kutta method in time t with the step size t. The errors of the numerical solution u_N are described by

We first use (3.6) to solve (3.5) numerically. The Hermite coefficients are calculated by the Hermite quadratures with N + 1 interpolation points. The errors E_N(t) and _N(t) at t = 1 with various values of N and t are listed in Tables 1 and 2, which show the high accuracy and the convergence of this method. Moreover the errors E_N(t) and _N(t) at various time with N = 8 and t = 0.001 are listed in Table 3, which indicates the stability of calculation. They coincide well with the theoretical analysis in the previous sections.

We next use (4.1) to solve (3.5). The corresponding errors (t) and (t) are defined in a similar way as for E_N(t) and _N(t). The errors (t) and (t) are presented in Tables 4-6. We find that scheme (3.6) provides the numerical results with the accuracy of the same order as (4.1). But the latter saves the work. Thus it is more preferable in actual computation.

As an another example, we take the test function

with b₁ = b₂ = 0.2 and h = 2.0. It decays algebraically and oscillates as x₁ and x₂ tend to the infinity. We also use (3.6) and (4.1) to solve (3.5) numerically as before. The corresponding errors (t) and (t) with various N and t are presented in Tables 7-9 for (3.6) and Tables 10-12 for (4.1). They also demonstrate the high accuracy, the covergence and the stability of both schemes.

6 Acknowledgment

The work of the first author was supported by The Science Foundation of Tongji university of China. The work of the second author was partially supported by The Special Funds for State Major Basic Research Subjects of China N. G1999032804, Shanghai Science Foundation N. 00JC14057 and The Special Funds for Major Speciality of Shanghai Education Committee.

Received: 28/XI/02.

#531/02.

• [1] D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia, (1977).
• [2] Y. Maday, B. Pernaud-Thomas and H. Vandeven, One réhabilitation des méthods spectrales de type Laguerre, Rech. Aérospat., 6 (1985), 353-379.
• [3] O. Coulaud, D. Funaro and O. Kavian, Laguerre spectral approximation of elliptic problems in exterior domains, Comp. Mech. in Appl. Mech and Engi., 80 (1990), 451-458.
• [4] D. Funaro, Estimates of Laguerre spectral projectors in Sobolev spaces, in Orthogonal Polynomials and Their Applications, ed. by C. Brezinski, L. Gori and A. Ronveaux, Scientific Publishing Co., (1991), 263-266.
• [5] Guo Ben-yu and Jie Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., 86 (2000), 635-654.
• [6] D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57 (1990), 597-619.
• [7] Guo Ben-yu, Error estimation for Hermite spectral method for nonlinear partial differential equations, Math. Comp., 68 (1999), 1067-1078.
• [8] Guo Ben-yu and Xu Cheng-long, Hermite pseudospectral method for nonlinear partial differential equations, RAIRO Math. Mdel. and Numer. Anal., 34 (2000), 859-872.
• [9] R.A. Adams, Sobolev Spaces, Academic Press, New York, (1975).
• [10] Guo Ben-yu, Spectral Methods and Their Applications, World Scientific, Singapore, (1998).
• [11] Guo Ben-yu, A class of difference schemes of two-dimensional viscous fluid flow, TR. SUST, 1965, Also see Acta Math. Sinica, 17 (1974), 242-258.
• [12] Guo Ben-yu, Generalized stability of discretization and its applications to numerical solution of nonlinear differential equations, Contemporary Math., 163 (1994), 33-54.
• [13] H.J. Stetter, Stability of nonlinear discretization algorithms, in Numerical Solutions of Partial Differential Equations, 111-123, ed. by J. Bramble, Academic Press, New York, (1966).

Publication Dates

• Publication in this collection
  20 July 2004
• Date of issue
  2003

History

• Received
  28 Nov 2002