Implicit explicit numerical schemes for Parabolic Integro-Differential Equations. Maya Briani. LUISS Guido Carli

Transcription

1 Implicit explicit numerical schemes for Parabolic Integro-Differential Equations Maya Briani LUISS Guido Carli & Istituto per le Applicazioni del Calcolo - CNR Joint work with R. Natalini (Istituto per le Applicazioni del Calcolo - CNR) G. Russo (Università di Catania)

2 Problem Numerical approximation of viscosity solutions for Parabolic Integro-Differential Equations (PIDE) t u L I (x, t, I, D, D 2 )u + H(x, t, Du, Iu) = 0 u(x, 0) = u 0 (x) C(R d ) L I linear degenerate elliptic operator Main application: Iu = R d M(u(x + z, t), u(x, t))µ x,t (dz) option pricing in jump-diffusion models Related models: radiative gas semiconductors H nonlinear first order operator 2

3 Plan Analytical results A convergence result Numerical implementation Implicit-Explicit methods 3

4 General framework t u L I (x, t, I, D, D 2 )u + H(x, t, Du, Iu) = 0 u(x, 0) = u 0 (x) u 0 C(R d ); L I linear degenerate elliptic operator; H nonlinear first order operator; Z Iu = M(u(x + z,t),u(x, t))µ x,t (dz); R d µ x,t positive bounded measure; M continuous function, such that: M(u,v) M(w, v) if u w; M(u,u) = 0; M(u,v) M(w, z) c((u w) + + v z ). A. L. Amadori (2000): Theorem of existence and uniqueness of viscosity solution A.L. Amadori, K. H. Karlsen, C. La Chioma, (2004): unbounded measure:! of viscosity solution 4

5 Numerical Approximation and Background F(x, t, u, Iu, Du, D 2 u) = L I (x, t, I, D, D 2 )u + H(x, t, Du, Iu) Problem t u + F(x, t, u, Iu, Du, D 2 u) = 0. Numerical grid in R d [0,T] h R d, k R: space, time grid steps. (x j,t n ) = (jh, nk), j Z d, n N, vj n = v(x j,t n ), ṽ = (vj n ) j,n. I h ṽ integral approximation. Scheme Q(h, k, j, n, v n j, I h ṽ, ṽ) = 0. 5

6 Properties and Hypotheses Q(h, k, j, n, v n j, I hṽ, ṽ) = 0 [H1] Stability [H2] Consistency [H3] Monotonicity of the approximating integral [H4] Monotonicity [H5] Comparison Principle (Amadori 2000) 6

7 Main Convergence Result Let assumption (H1) (H5) hold true. Then, as (h, k) 0, the numerical solution ṽ converges locally uniformly to the unique continuous viscosity solution. M. Briani, C. La Chioma, R. Natalini - Numerische Mathematik, August 2004 (on line version) Convergence - purely second order problems (i.e. without integral term): Barles, Souganidis (1991). 7

8 Numerical Algorithms Notice that: Q(h, k, j, n, v n j, ṽ) verifying the convergence (differential) conditions of stability, consistency and monotonicity and Q(h, k, j, n, v n j, I h ṽ, ṽ) = Q(h, k, j, n, v n j, ṽ) I h ṽ, monotonicity of the integral approximation if v n j = w n j and ṽ w, then I h ṽ I h w ( the weights of the integral approximation are greater than zero!) convergence Example: An Explicit Finite differences scheme coupled with quadrature rule for the integral term... but: under CFL condition!!! 8

9 Time approximation u t (x, t) + H(Du(x, t), Iu(x, t)) = G(u(x, t), D 2 u(x, t)) Euler forward: u n+1 j k u n j + H(u n j, I h u n j ) = G(u n j ) The scheme is second order in time but stable under the parabolic CFL condition: k ch 2 h = 1/100 k = 1/10000!!! Euler backward: u n+1 j k u n j + H(u n+1 j, I h u n+1 j ) = G(u n+1 j ) The scheme is unconditionally stable but not practically feasible: I h ũ n+1 dense system!!! 9

10 IMEX (Implicit-Explicit)technique IMEX technique has been introduced for time dependent partial-differential equations that involve terms of different types. IMEX schemes have been widely used for the time integration of spatially discretized PDEs of diffusion-convection type. Some schemes of this type were proposed and analyzed as far back as the late 1970 s. Istances of these methods have been successfully applied to the Navier-Stokes equations... For recent developments: L. Pareschi and G. Russo (2003) 10

11 IMEX (Implicit-Explicit) scheme u t (x, t) + H(Du(x, t), Iu(x, t)) = G(u(x, t), D 2 u(x, t)) Implicit in G(u(x, t), D 2 u(x, t)) To avoid parabolic CFL condition To avoid dense system Explicit in H(Du(x, t), Iu(x, t)) To get second or hight order accuracy in time M. Briani, G. Russo, R. Natalini; Implicit-Explicit numerical scheme for integro-differential parabolic problems arising in financial theory, IAC report 38 (2004). 11

12 IMEX (Implicit-Explicit) scheme u t (x, t) + H(Du(x, t), Iu(x, t)) = G(u(x, t), D 2 u(x, t)). Time approximation, for i = 1,..., ν ν u (i) = u n k ã ij H(Du (j), Iu (j) ) + k j=1 ν a ij G(u (j), D 2 u (j) ) j=1 u n+1 = u n k ν ω i H(Du (i), Iu (i) ) + k ν ω i G(u (i), D 2 u (i) ) i=1 i=1 The matrices à = (ã ij ), ã ij = 0 for j i and A = (a ij ) are ν ν matrices such that the resulting scheme is explicit in H and implicit in G. 12

13 IMEX (Implicit-Explicit) scheme The Midpoint(1, 2, 2) scheme 8 u >< (2) = u n k 2 H(Dun, Iu n ) + k 2 G(u(2), D 2 u (2) ) >: u n+1 = u n kh(du (2), Iu (2) ) + kg(u (2), D 2 u (2) ) A two-stage, third-order scheme 8 u (2) = u n kγh(du n, Iu n ) + kγg(u (2), D 2 u (2) ) >< >: u (3) = u n k(γ 1)H(Du n, Iu n ) 2k(γ 1)H(Du (2), Iu (2) ) +k(1 2γ)G(u (2),D 2 u (2) ) + kγg(u (3),D 2 u (3) ) u n+1 = u n k «H(Du (2), Iu (2) ) + H(Du (3), Iu (3) ) 2 + k «G(u (2), D 2 u (2) ) + G(u (3),D 2 u (3) ) 2 13

14 Stability and Order Accuracy The Midpoint scheme is of second order in time and stable under the CFL condition k ch 4/3 The third-order (3Ord) scheme is of third order in time and stable under the CFL condition k ch 1 14

15 Numerical test u (x, t) + a u t x (x, t) = u b 2 (x, t) + Iu(x, t) (x, t) R [0, T] x2 u(x, 0) = u 0 (x) with Iu = + K 0 and [u(x + z, t) u(x, t)]k(z)dz R K(x)dx = 1. option pricing - Merton Model u 0 (x) = (e x E) +, (call option); K(x) = 1 δ 2π exp( x2 2δ 2) 15

16 Numerical Implementation Truncation of the problem domain R [a, b] R Truncation of the integral domain + +zɛ z ɛ Difficulty: nonlocal nature of the integral term + convolution boundary conditions 16

17 Numerical Implementation The numerical domain v n+1 j = ( kw 1 vj 1 n + 1 kw 0 )v j n + kw 1vj+1 n j 1 j j+1 17

18 Numerical Implementation The numerical domain p +λkh α i vj+i(γ n δ ) i i= p j p j j+p 18

19 Numerical Implementation +λkh p The numerical domain i= p α i v n j+i(γ δ ) i + λkh i>p α i? v n j+i (Γ δ ) i, n+1 n a j p j b j+p 19

20 Numerical Implementation boundary conditions The diffusive effect of the integral operator: Out of the numerical domain [a, b] u v If δ 1, v solves v t + av x bv xx + cv = λδ2 2 v xx u v L (0,T;L 1 (R)) O(Tδ 3 ). 20

21 Computational costs IMEX-DIRK scheme time space integral computational cost N P 2 or (3P log 2 (P) + P) + M... = O(MN(3P log 2 (P) + P)) + N P 2 or (3P log 2 (P) + P) The explicit approximation time space integral computational cost M N P 2 or (3P log 2 (P) + P) = O(MN(3P log 2 (P) + P)) 21

22 CPU Times N explicit scheme Midpoint-122 ARS-233 SSP s 0.04s 0.04s 0.07s s 0.35s 0.32s 0.24s s 3.21s 2.33s 1.73s m56,6s 29.58s 17.62s 13.48s Table 1: CPU times on 1.6 GHz Pentium IV PC when T = 1. 22

23 Errors and Convergence Orders ( e h ) 1, γ 1, = log 2 e h/2 1, e h 1 = h j v h j (T) v h/2 2j (T) eh = v h (x, T) v h/2 (x, T) T = 5 Midpoint-122 h k e 1 γ 1 e γ

24 Errors and Convergence Orders T = 5 ARS-233 h k e 1 γ 1 e γ T = 5 SSP3-433 h k e 1 γ 1 e γ

25 Conclusions Rigorous convergent result Fast and accurate finite difference schemes for linear parabolic integro-differential equations Further developments Non-linear problems Higher order approximation in space... M. Briani, C. La Chioma, R. Natalini; Numerische Mathematik - August 2004 (on line version). M. Briani, G. Russo, R. Natalini; Implicit-Explicit numerical scheme for integro-differential parabolic problems arising in financial theory. 25