Qualitative Properties of Numerical Approximations of the Heat Equation

Transcription

1 Qualitative Properties of Numerical Approximations of the Heat Equation Liviu Ignat Universidad Autónoma de Madrid, Spain Santiago de Compostela, 21 July 2005

2 The Heat Equation { ut u = 0 x R, t > 0, u(0, x) = ϕ(x) x R. (1) Decay Properties of the Linear Semigroup e t Theorem 1. Let 1 q p. Then there exists a positive constant c(p, q) such that e t ϕ L p (R) c(p, q)t 1 2 (1 q 1 p ) ϕ L q (R). (2) p = q p =, q = 1

3 The solution e t ϕ = G(t) ϕ The Fundamental Solution G(t, x) = (4πt) 1/2 exp ( x 2 4t ) Time Decay of G(t, x) G(t) L p (R) C pt 1 2 (1 1 p )

4 Consider the finite difference approximation du h dt hu h = 0, t > 0, u h (0) = ϕ h. (3) Here u h stands for the infinite vector unknown {u h j } j Z,, u j(t) being the approximation of the solution at the node x j = jh, and h being the classical second order finite difference approximation of 2 x : ( h u) j = 1 h 2[u j+1 2u j + u j 1 ]. Has the numerical approximation (8) the same decay rates as the continuous model?

5 Motivation: There are numerical schemes based numerical viscosity Example: To recover the dispersive properties of the Schrödinger Equation we add numerical viscosity : L. Ignat and E. Zuazua, CRAS, , 2005 The simpler scheme is replaced by i duh dt + hu h = 0, t > 0, u h (0) = ϕ h. (4)

6 i duh dt + hu h = ia(h) h u h, t > 0, u h (0) = ϕ h, where a(h) > 0 is such that a(h) 0 as h 0. Remark: The solution is a combination of the linear semigroups generated by (8) and (4).

7 Preliminaries For 1 p < we define the spaces l p h (Z) as l p h (Z) = {(ϕ j) j Z C Z : h ϕ j p < }. j Z l p h (Z, x m ) = {{ϕ j } j Z : h jh pm ϕ j p < } j Z The discrete convolution ((a b) j ) j Z of two sequences (a j ) j Z and (b j ) j Z is defined by (a b) j = h a j n b n = h a n b j n n Z n Z

8 For v lh 2 (Z) we define the semidiscrete Fourier transform ˆv(ξ) = h [ e ijhξ v j, ξ π h, π ] h j Z v j = 1 2π π/h π/h eijhξˆv(ξ)dξ, j Z. ˆv L 2 ( π/h,π/h) = 2π v l 2 h (Z).

9 Back to the Semidiscrete Heat Equation du h dt hu h = 0, t > 0, u h (0) = ϕ h. Exact solution u h (t) = K h t ϕh Fundamental Solution : K h t Kt h 4t (ξ) = e h 2 sin2 ( ξh 2 ), ξ [ π h, π ] h (K h t ) j = 1 2π π h π h e 4t h 2 sin2 ( ξh 2 ) e ijhξ dξ

10 Decay Properties Decay Rates of Fundamental Solution Theorem 2. If 1 p, then there is a positive constant C(p) independent of h, such that K h t l p h (Z) C(p)t 1 2 (1 1 p ), t > 0. (5) Proof Ideas: Interpolation between the cases p = 1 and p =. p = : 4 h 2 sin2 ( ξh 2 ) ξ 2, ξ [ π h, π ] h p = 1: Lemma 1. (Carlson - Beurling Inequality) Assume that a L 2 (R), a L 2 (R). Then â L 1 (R) and â L 1 (R) (2 a L 2 (R) a L 2 (R) )1 2 (6)

11 More about the fundamental solution K h t (K h t ) j = e 2t h 2 πh I j ( ) 2t h 2 I ν (x) is the modified Bessel function Using the properties of Bessel s function we prove Lemma 2. For any positive numbers h, t and for all integers j, (K h t ) j is positive. Also the map j (K h t ) j is increasing for j 0 and decreasing for j 0.

12 Notation : the discrete gradient ( + h f) j = f j+1 f j h Theorem 3. If 1 p, then there is a positive constant C(p) independent of h, such that + h Kh t l p h (Z) C(p)t 1 2 (1 1 p ), t > 0. (7)

13 A finer Analysis of the long time behavior Zuazua and Duandikoethea (C.R.Acad. Sci. Paris, 1992, ) show that if u(x, t) is solution of the Heat Equation with initial data ϕ L 1 (R, 1 + x m+1 ) then u(, t) where 0 α m ( 1) α α! ( ϕ(x)x α dx)d α G(, t) L p t 1 2 (m+2 1 p ) ϕ L 1 ( x m+1 ) G(x, t) = (4πt) 1/2 exp( x 2 4t ) is the fundamental solution of the heat equation.

14 What we can say about the semidiscrete case? du h dt hu h = 0, t > 0, u h (0) = ϕ h. (8)

15 A Discrete Result There is a function K h (t, x) such that uh (t) m α=0 ( 1) α M α ϕ h α! α K h (t, ) x α l q (hz) (9) C(m)t 1 2 (m+2 1 q ) ϕ h l 1 (hz, x m+1 ) for all ϕ h l 1 (hz, x m+1 ) and t > 0, uniformly in h > 0. M 0 ϕ h = h n Z ϕ h n M α ϕ h = h n Z (nh) α ϕ h n, α 1

16 More about the function K h t (x) K h (t, x) = 1 2π π h π h e 4t h 2 sin2 ( θh 2 ) e ixθ dθ. (10) K h (t, jh) = (K h t ) j Theorem 4. For all m N and 1 p there is a constant C(m, p) > 0 such that m K h (t, ) x m for all h > 0 and t > 0. L p (R) C(m, p)t (m p ) (11)

17 Proof of The Discrete Result (u h (t)) j = (K h t ϕh ) j = h n Z K h (t, (j n)h)ϕ h n = h u h (t) ϕ h n K t h n 0 m α=0 ( 1) α M α ϕ h n α! ((j n)h) m = h n 0 α=0 ϕ h n b jn α K h t x α j = (nh) α α Kt h α! x α (jh)

18 It suffices to consider the cases q =, q = 1 Estimates for b jn b jn nh m+1 m! m+1 K h (t, ) x m+1 L (R) b jn nh m m! max{jh,(j n)h} min{jh,(j n)h} m+1 K h (t, ) x m+1 dx

19 The Convection-Diffusion Equation { ut u = a ( u q 1 u) in (0, ) R N u(0) = u 0 (12) Escobedo, Zuazua, J. Func. Analysis, 1991 : Global Existence and Uniqueness for u 0 L 1 (R N ) q > N, u 0 L 1 (R N, 1 + x ) L (R N ) the large time behavior is given by the heat kernel

20 u(t) ( ) u 0 G(t) t N 2 (1 1 p ) a(t), a(t) 0, t, p [1, ] p

21 A Semidiscrete Scheme du h j = uh j+1 2uh j + uh j 1 dt h 2 + uh j+1 q 1uh j+1 uh j q 1uh j, t > 0, j Z h u h j (0) = ϕh j, j Z. (13) Theorem 5. For any initial data u 0 lh 1 (Z) there exists a unique solution u h C 1 ([0, ); l 1 h (Z)) of the problem (13) which satisfies u h (t) l 1 h (Z) uh 0 l 1 h (Z), uh (t) l h (Z) uh 0 l h (Z). (14)

22 Decay of Solutions Theorem 6. For every p [1, ] there exists some constants C(p) such that u h (t) l p h (Z) C(p)t 1 2 (1 1 p ) ϕ h l 1 h (Z), (15) for all t > 0, uniformly on h > 0. Proof. d h u h j dt p p( + h uh ) ( + h uh p 2 u h ) c(p) + h ( uh 2) p 2 l h 2 j Z (Z). (16) + Discrete Sobolev Inequalities, Gronwall s Inequality

23 Asymptotic Behavior Let q > 2, u h 0 l1 h (Z, 1+ x ) l h (Z) and M 0 = h j Z (uh 0 ) j. Then uniformly on h > 0. u h (t) M 0 K h t l p h (Z) C pt 1 2 (1 1 p ) a(t), t > 0 a(t) = t 1/2 q > 3 t 1/2 log(t + 2) q = 3 t (q 2)/2 2 < q < 3

24 Proof : u h (t) = K h t u h 0 + t 0 + K h t s u h (s) q 1 u h (s)ds First step K h t uh 0 M 0K h t l p h (Z) C uh 0 l 1 h (Z, x )t 1 2 (1 1 p ) 1 2 by using estimates for the solutions of the discrete heat equation (9)

25 Second step t 0 + h Kh t s uh (s) q 1 u h (s)ds l p h (Z) C( uh 0 l 1 h (Z), uh 0 l h (Z))t 1 2 (1 1 p ) a(t) estimations for + h Kh t (7) a priori estimates for u h (15)

26 The same results can be proved in the case of du h j = ( 1) m 1 ( m dt h u) j, t > 0, j Z d u h j (0) = ϕh j, j Zd, (17) Future Work Full discrete schemes The long time behavior for other nonlinear problems du h j dt = ( 1)m 1 ( m h u) j + f(u h )