# An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

Authors: Lei Ren; Lei Liu
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.


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## Body

## 1. Introduction

The Black-Scholes model, proposed in 1973 by Black and Scholes [1] and Merton [2], gives a theoretical estimate of the price of European-style options. Until Now, some of Black-Scholes models involving the fractional derivatives have emerged. In [3], Wyss priced a European call option by a time-fractional Black-Scholes model. In [4], Liang et al. derive a biparameter fractional Black-Merton-Scholes equation and obtain the explicit option pricing formulas for the European call option and put option, individually. An explicit closed-form analytical solution for barrier options under a generalized time-fractional Black-Scholes model by using eigenfunction expansion method together with the Laplace transform is derived in [5]. In [6], a discrete implicit numerical scheme with a spatially second-order accuracy and a temporally 2-α order accuracy is constructed; the stability and convergence of the proposed numerical scheme are analysed using Fourier analysis. In [7], H.Zhang et al. use some numerical technique to price a European double-knock-out barrier option, and then the characteristics of the three fractional Black-Scholes models are analysed through comparison with the classical Black-Scholes model. More recently, a numerical scheme of fourth-order in space and 2-α in time is derived in [8]; the solvability and convergence of the proposed numerical scheme are proved rigorously using a Fourier analysis. Some computationally efficient numerical methods have been proposed for solving fractional differential equation, for example, which include finite difference methods, finite element methods, finite volume methods, spectral methods, and meshless methods [9–26].In this paper, we continue the work of R.H.De Staelen et al. [8]. The class of equations is given by(1)∂αCS,t∂tα+12σ2S2∂2CS,t∂S2+r-DS∂CS,t∂S=rCS,t,S,t∈Bd,Bu×0,Twith the following boundary (barrier) and final conditions(2)CBd,t=Pt,CBu,t=Qt,t∈0,T,and its initial condition(3)CS,T=VS,S∈Bd,Bu,where r is the risk free rate, D is the dividend rate, and σ>0 is the volatility of the returns. The functions P and Q are the rebates paid when the corresponding barrier is hit. The terminal playoff of the option is V(S). The fractional derivative in (1) is a Caputo derivative defined as(4)∂αCS,t∂tα=1Γ1-α∫0t∂C∂ηS,ηt-η-αdη,0<α<1,∂CS,t∂t,α=1.As described in [8], we consider the transform problem of (1)(5)D0CtαUx,t=a∂2Ux,t∂x2+b∂Ux,t∂x-cUx,t+fx,t,x,t∈0,∞×0,T,Ubd,t=pt,Ubu,t=qt,t∈0,T,Ux,0=φx,x∈bd,bu.The rest of the paper is organized as follows: in Section 2, an efficient implicit numerical scheme with second-order accuracy in time and fourth-order accuracy in space is constructed. The analysis of the stability and convergence are presented in Section 3. In Section 4, numerical examples are given to illustrate the accuracy of the presented scheme and to support our theoretical results. Concluding remarks are given in the last section.


## 2. Construction of the Compact Finite Difference Scheme

In order to simplify the computation and analysis of the following compact finite difference scheme for Black-Scholes model, we use an indirect approach by introducing a suitable transformation.According to some simple calculations, we transform equation (5) into(6)D0CtαVx,t=a∂2Vx,t∂x2-dVx,t+gx,t,x,t∈0,∞×0,T,Vbd,t=p∗t,Vbu,t=q∗t,t∈0,T,Vx,0=φ∗x,x∈bd,bu.where(7)p∗t=pt,q∗t=kbuqt,φ∗x=kxφx.It is clear that U(x,t) is a solution of (5) if and only if V(x,t) is a solution of (6).In order to construct the compact finite difference scheme for the problem (5), we consider the above equivalent form (6).Letτ=T/N be the time step and h=bu-bd/M=L/M be the spatial step, where M,N are positive integers.Since the grid functionv={vi∣0≤i≤M}, we then define difference operators as follows: (8)δxvi-1/2=1hvi-vi-1,δx2vi=1h2vi+1-2vi+vi-1,Hxvi=vi+h212δx2vi,We also define(9)a0=σ1-α,b0=0,ak=k+σ1-α-k-α21-αk≥1,bk=12-αk+σ2-α-k-α22-α-12k+σ1-α+k-α21-αk≥1. where σ=1-α/2, and(10)ck,n=a0,k=0,n=1ak+bk+1-bk,0≤k≤n-2,n≥2an-1-bn-1,k=n-1,n≥2.Lemma 1.
It holds (see [27])(11)1-α2k+σ-α<ak-bk<k+σ1-α-k-α21-αk≥1,In order to discretize (6) into a compact finite difference system, we introduce the following lemmas.Lemma 2.
Assumingv(t)∈C3[0,T], we have(12)D0Ctαvtn-α/2=1μ∑k=1ncn-k,nvtk-vtk-1+Oτ3-α.where μ=ταΓ(2-α).Proof.
From Lemma 2 of [9], we can obtain the proof of lemma.Lemma 3.
Assumingv(t)∈C2[0,T]. When n≥1, we obtain(13)vtn-α/2=α2vtn-1+1-α2vtn+Oτ2.Proof.
According to some simple calculations, the proof follows from Taylor expansions of the functionv(t) at the point tn-α/2 for t=tn-1 and t=tn.Since the above lemmas, we then discretize (6) into a compact finite difference scheme. In order to analyse, we define (14)δtvn-1/2=1τvn-vn-11≤n≤N,vn,α/2=α2vn-1+1-α2vn1≤n≤N. We also define the grid functions as follows: (15)Vin=Vxi,tn,Win=∂Vxi,tn∂t,Zin=∂2Vxi,tn∂x2,gin=gxi,tn,gin-α/2=gxi,tn-α/2,p∗,n=p∗tn,q∗,n=q∗tn,φi∗=φ∗xi. For the second-order spatial derivative Zin, we adopt the following fourth-order compact approximation (see [28])(16)HxZin=δx2vxi+Oh4,1≤i≤M-1,We consider equation (6) at the point (xi,tn-α/2); we can obtain(17)D0CtαVxi,tn-α/2=aZin-α/2-dVin-α/2+gin-α/2.From Lemmas2 and 3, we have(18)1μ∑k=1ncn-k,nVik-Vik-1=aZin,α/2-dVin,α/2+gin-α/2+Rtαin,0≤i≤M,1≤n≤N,where(19)Rtαin=a-dOτ2+Oτ3-α,0≤i≤M,1≤n≤N.We apply Hx to equation (18); then we have(20)1μ∑k=1ncn-k,nHxVik-Vik-1=aδx2Vin,α/2-dHxVin,α/2+Hxgin-α/2+Rtxαin,1≤i≤M-1,1≤n≤N,where(21)Rtxαin=HxRtαin+aRxαin,1≤i≤M-1,1≤n≤Nand(22)Rtxαin≤CRτ2+h4,1≤i≤M-1,1≤n≤N.If we omit (Rtxα)in, then we have the compact finite difference scheme:(23)1μ∑k=1ncn-k,nHxvik-vik-1=aδx2vin,α/2-dHxvin,α/2+Hxgin-α/2,1≤i≤M-1,1≤n≤N,vbd,t=p∗t,vbu,t=q∗t,t∈0,T,vx,0=φ∗x,x∈bd,bu.


## 3. Stability and Convergence of the Proposed Compact Difference Scheme

Theorem 4.
The compact difference scheme (23) is uniquely solvable.Proof.
The compact difference scheme (23) can be written in matrix form (24)AVn=bn-1, where (25)bn-1=∑k=0n-1ζkVk,ζk∈R The tridiagonal coefficient matrix A=(aij) yields (26)aii=56a0α+b1αμ+a2-αh2+52-α12d,∑j≠iaij=16a0α+b1αμ-a2-αh2+2-α12d.It is easy to see that the tridiagonal coefficient matrix A is strictly diagonally dominant. Therefore, the coefficient matrix is nonsingular and hence invertible.Next, we consider the stability and convergence analysis of the compact difference scheme (23).LettingΩ={u∣u=(u0,u1,…,uM),u0=uM=0}, for grid functions u,v∈Ω, we define the inner product and norm as follows: (27)u,v=h∑i=1M-1uivi,u=u,u1/2,u∞=max0≤i≤Mui.δxu,δxv=h∑i=0M-1δxui+1/2δxvi+1/2,u1=δxu,δxu1/2,u1=u2+u121/2.According to simple calculations, we obtain(28)δx2u,v=-δxu,δxv,hδx2u≤2u1,hu1≤2u.In order to analyse, we introduce the discrete inner product and norm:(29)u,v=Hxu,-δx2v=δxu,δxv-h212δx2u,δx2v,uε=u,u1/2. Based on above inner product and norm, we have the following lemmas.Lemma 5 (see [29]).
Supposeu∈Ω, we obtain(30)Hxu2≤u2≤3L216uε2,u∞2≤3L8uε2,u12≤38+L216uε2.Lemma 6 (see [27]).
Supposeu∈Ω, we obtain(31)u≤L28δx2u,uε2≤3L216δx2u2.Lemma 7 (see [9]).
Supposeu∈Ω, we obtain(32)∑k=1ncn-k,nHxuk-uk-1,-δx2un,α/2≥12∑k=1ncn-k,nukε2-uk-1ε2,1≤n≤N.In the next, we then analyse the stability and convergence of the scheme (23).Theorem 8 (stability).
Letvn=(v0n,v1n,…,vMn) be the solution of the compact difference scheme (23) with v0n=vMn=0. Assume that one of the conditions 1≤4(4ε-1)a/3dεL2 holds for some positive constant ε>1/4.
Then it holds(33)vnε2≤v0ε2+4εΓ1-αTαamax1≤n≤NHxgn-α/22,1≤n≤N.Proof.
We take the inner product of equation (23) with -δx2vn,α/2 yield (34)1μ∑k=1ncn-k,nHxvk-vk-1,-δx2vn,α/2=-aδx2vn,α/22-dHxvn,α/2,δx2vn,α/2-Hxgn-α/2,δx2vn,α/2,1≤n≤N. Using Lemma 7,(35)12μ∑k=1ncn-k,nvkε2-vk-1ε2≤-aδx2vn,α/22+dvn,α/2ε2-Hxgn-α/2,δx2un,α/2,1≤n≤N.When 1≤4(4ε-1)a/3dεL2 for some positive constant ε>1/4, we have from the Cauchy-Schwarz inequality and Lemmas 6 that(36)dvn,α/2ε2≤3dL216δx2vn,α/22≤a-a4εδx2vn,α/22(37)-Hxgn-α/2,δx2un,α/2≤εaHxgn-α/22+a4εδx2vn,α/22 By (35) and the Cauchy-Schwarz inequality,(38)-aδx2un,α/22+dvn,α/2ε2-Hxgn-α/2,δx2un,α/2≤εaHxgn-α/22.Substituting (38) into (35) leads to(39)∑k=1ncn-k,nukε2-uk-1ε2≤2εμaHxgn-α/22. The above inequality can be rewritten as (40)c0,nunε2≤∑k=1n-1cn-k-1,n-cn-k,nukε2+cn-1,nu0∗2+2εμaHxgn-α/22.Since by the definition of cn-1,n, (41)μcn-1,n=μan-1-bn-1<2Γ1-αTα, we have from (40) that(42)c0,nunε2≤∑k=1n-1cn-k-1,n-cn-k,nukε2+cn-1,nu0ε2+4εΓ1-αTαaHxgn-α/22.Letting(43)E=u0ε2+4εΓ1-αTαamax1≤n≤NHxgn-α/22and assuming ukε2≤E(0≤k≤n-1), we obtain(44)c0,nunε2≤∑k=1n-1cn-k-1,n-cn-k,nE+cn-1,nE=c0,nE.and we have the needed estimates.Lettingein=Vin-vin, we get the following error equation:(45)1μ∑k=1ncn-k,nHxeik-eik-1=aδx2ein,α/2-dHxein,α/2+Rtxαin,1≤i≤M-1,1≤n≤N,ebd,t=0,ebu,t=0,t∈0,T,ex,0=0,x∈bd,bu.Since the above error equation (45), we now obtain the following convergence results.Theorem 9 (convergence).
LetVin denote the value of the solution v(x,t) of (23) at the mesh point (xi,tn) and let vn=(v0n,v1n,…,vMn) be the solution of the compact difference scheme (23). Then when 1≤4(4ε-1)a/3dεL2, it holds(46)Un-unε≤C1τ2+h4,1≤n≤N,where(47)C1=4Γ1-αTαLCR2a1/2,Proof.
It follows from Theorem8 that (48)enε2≤4εΓ1-αTαamax1≤n≤NRtxαin2,1≤n≤N, Applying (22), we get (49)enε2≤C12τ2+h42. The estimate (46) is proved.Remark 10.
The constraint condition1≤4(4ε-1)a/3dεL2 in Theorems 8 and 9 is only for the analysis of the stability and convergence of the compact difference scheme (23). This condition is easily verifiable for practical problems.


## 4. Numerical Experiment

For demonstrating the efficiency of the compact difference scheme (23), we make two numerical experiments of it.SupposeVin=v(xi,tn) be the value of the solution v(x,t) of the problem (1)–(3) at the mesh point (xi,tn). From (22), we can see that(50)Vn-vnν≤C2τ2+h4,ν=1,2,∞where C2 is a positive constant independent. In order to check this accuracy of the compact difference scheme, we compute the following norm errors:(51)Eντ,h=max0≤n≤NVn-vnνν=1,2,∞.The temporal convergence order and the spatial convergence order are denoted by(52)Oνtτ,h=log2Eν2τ,hEντ,h,Oνsτ,h=log2Eντ,2hEντ,hν=1,2,∞.Example 1.
We first consider a problem, which is governed by equation (1) in [0,1]×[0,1] with r=0.05,σ=0.25,a=σ2/2,b=r-a,c=r and(53)fx,t=2t2-αΓ3-α+2t1-αΓ2-αx21-x-t+12a2-6x+b2x-3x2-cx21-x.The boundary and initial conditions are given by (2) and (3) with(54)Ux,0=x21-x,U0,t=U1,t=0.It is easy to check that U(x,t)=(t+1)2x2(1-x) is the solution of this problem.
For differentα, we let the spatial step h=1/100. Table 1 gives the errors Eν(τ,h)(ν=1,2,∞) and the temporal convergence orders Oνt(τ,h)(ν=1,2,∞) of the computed solution Uin for α=1/4,1/2,3/4 and different time step τ. From the table, we can see that the computed solution Uin has the second-order temporal accuracy. For comparison, the corresponding temporal convergence orders Oνt(τ,h)(ν=∞) given in [8] has only 2-α order; thus it is far less accurate than the compact difference scheme (23) given in this paper.
Next, we compute the spatial convergence order of the compact difference scheme (23). Table 2 presents the errors Eν(τ,h)(ν=1,2,∞) and the spatial convergence orders orderOνs(τ,h)(ν=1,2,∞). The table demonstrates that the compact difference scheme (23) has the fourth-order spatial accuracy.Table 1
The errors and the temporal convergence orders of the compact difference scheme (23) for Example 1  (h=1/100).
α τ E 1 ( τ , h ) O 1 t ( τ , h ) E 2 ( τ , h ) O 2 t ( τ , h ) E ∞ ( τ , h ) O ∞ t ( τ , h ) 1/4 1/10 3.7361e–05  3.7314e–05  6.2608e–05 1/20 9.3611e–06 1.9968 9.3492e–06 1.9968 1.5684e–05 1.9970 1/40 2.3429e–06 1.9984 2.3399e–06 1.9984 3.9251e–06 1.9985 1/80 5.8603e–07 1.9992 5.8528e–07 1.9992 9.8176e–07 1.9993 1/160 1.4654e–07 1.9997 1.4635e–07 1.9997 2.4549e–07 1.9997 1/320 3.6634e–08 2.0001 3.6587e–08 2.0001 6.1372e–08 2.0000  1/2 1/10 6.7788e–05  6.7702e–05  1.1393e–04 1/20 1.6994e–05 1.9960 1.6972e–05 1.9960 2.8555e–05 1.9964 1/40 4.2543e–06 1.9980 4.2489e–06 1.9980 7.1480e–06 1.9981 1/80 1.0643e–06 1.9990 1.0630e–06 1.9990 1.7882e–06 1.9991 1/160 2.6617e–07 1.9995 2.6583e–07 1.9995 4.4718e–07 1.9996 1/320 6.6548e–08 1.9999 6.6463e–08 1.9999 1.1181e–07 1.9999  3/4 1/10 8.8226e–05  8.8110e–05  1.4950e–04 1/20 2.2098e–05 1.9973 2.2069e–05 1.9973 3.7435e–05 1.9976 1/40 5.5299e–06 1.9986 5.5226e–06 1.9986 9.3672e–06 1.9987 1/80 1.3832e–06 1.9993 1.3813e–06 1.9993 2.3429e–06 1.9993 1/160 3.4587e–07 1.9996 3.4542e–07 1.9996 5.8585e–07 1.9997 1/320 8.6475e–08 1.9999 8.6361e–08 1.9999 1.4647e–07 1.9999Table 2
The errors and the spatial convergence orders of the compact difference scheme (23) for Example 1  (h=1/10000).
α τ E 1 ( τ , h ) O 1 t ( τ , h ) E 2 ( τ , h ) O 2 t ( τ , h ) E ∞ ( τ , h ) O ∞ t ( τ , h ) 1/4 1/2 1.1190e–04  6.4607e–05  9.1369e–05 1/4 5.4155e–06 4.3690 4.2667e–06 3.9205 5.4429e–06 4.0693 1/8 2.9041e–07 4.2209 2.6840e–07 3.9907 3.3922e–07 4.0041 1/16 1.7125e–08 4.0840 1.6758e–08 4.0015 2.1165e–08 4.0024 1/32 1.0264e–09 4.0604 1.0207e–09 4.0372 1.2889e–09 4.0375  1/2 1/2 1.0340e–04  5.9701e–05  8.4430e–05 1/4 5.0151e–06 4.3659 3.9472e–06 3.9189 4.9995e–06 4.0779 1/8 2.6907e–07 4.2202 2.4841e–07 3.9900 3.1142e–07 4.0048 1/16 1.5831e–08 4.0871 1.5487e–08 4.0036 1.9400e–08 4.0047 1/32 9.2534e–10 4.0967 9.2011e–10 4.0731 1.1538e–09 4.0716  3/4 1/2 9.3459e–05  5.3959e–05  7.6309e–05 1/4 4.5477e–06 4.3611 3.5734e–06 3.9165 4.4770e–06 4.0912 1/8 2.4420e–07 4.2190 2.2506e–07 3.9889 2.7865e–07 4.0060 1/16 1.4335e–08 4.0905 1.4014e–08 4.0053 1.7372e–08 4.0036 1/32 8.2043e–10 4.1270 8.1562e–10 4.1028 1.0150e–09 4.0971Example 2.
In this example, we test the error and the convergence order of the compact difference scheme (23). Consider equation (1) in the domain [0,1]×[0,1] with r=0.5,a=1,b=r-a,c=r and(55)fx,t=2t2-αΓ3-α+2t1-αΓ2-αx3+x2+1-t+12a2+6x+b2x+3x2-cx3+x2+1.The boundary and initial conditions are given by (2) and (3) with(56)ϕ0t=x3+x2+1,U0,t=t+12,U1,t=3t+12.It is clear that U(x,t)=(t+1)2(x3+x2+1) is the exact analytical solution of this problem.
Apply the compact difference scheme (23) to solve the above problem. Table 3 presents the errors Eν(τ,h)(ν=1,2,∞) and the temporal convergence orders Oνt(τ,h)(ν=1,2,∞); we can see that the computed solution Uin has the second-order temporal accuracy.
From Table4, we can obtain the errors Eν(τ,h)(ν=1,2,∞) and the spatial convergence orders Oνs(τ,h)(ν=1,2,∞). These numerical results demonstrate that the accuracy of the compact difference scheme (23) is fourth-order.Table 3
The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2  (τ=1/100).
α h E 1 ( τ , h ) O 1 s ( τ , h ) E 2 ( τ , h ) O2s(τ,h) E ∞ ( τ , h ) O ∞ s ( τ , h ) 1/4 1/10 9.5681e–05  9.5622e–05  1.3266e–04 1/20 2.3985e–05 1.9961 2.3971e–05 1.9961 3.3254e–05 1.9961 1/40 6.0044e–06 1.9981 6.0007e–06 1.9981 8.3244e–06 1.9981 1/80 1.5021e–06 1.9990 1.5012e–06 1.9990 2.0825e–06 1.9991 1/160 3.7564e–07 1.9996 3.7541e–07 1.9996 5.2078e–07 1.9996 1/320 9.3916e–08 1.9999 9.3859e–08 1.9999 1.3020e–07 1.9999  1/2 1/10 1.7283e–04  1.7272e–04  2.4033e–04 1/20 4.3358e–05 1.9950 4.3331e–05 1.9950 6.0286e–05 1.9951 1/40 1.0858e–05 1.9975 1.0852e–05 1.9975 1.5097e–05 1.9976 1/80 2.7169e–06 1.9987 2.7153e–06 1.9987 3.7774e–06 1.9988 1/160 6.7952e–07 1.9994 6.7910e–07 1.9994 9.4474e–07 1.9994 1/320 1.6991e–07 1.9998 1.6980e–07 1.9998 2.3622e–07 1.9998  3/4 1/10 2.2075e–04  2.2061e–04  3.0894e–04 1/20 5.5326e–05 1.9964 5.5291e–05 1.9964 7.7418e–05 1.9966 1/40 1.3849e–05 1.9981 1.3841e–05 1.9981 1.9378e–05 1.9983 1/80 3.4646e–06 1.9991 3.4624e–06 1.9991 4.8476e–06 1.9991 1/160 8.6641e–07 1.9995 8.6587e–07 1.9995 1.2123e–06 1.9996 1/320 2.1663e–07 1.9998 2.1650e–07 1.9998 3.0311e–07 1.9998Table 4
The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2  (h=1/15000).
α τ E 1 ( τ , h ) O 1 t ( τ , h ) E 2 ( τ , h ) O 2 t ( τ , h ) E ∞ ( τ , h ) O ∞ t ( τ , h ) 1/4 1/2 1.2909e–04  7.4532e–05  1.0540e–04 1/4 6.2469e–06 4.3691 4.9219e–06 3.9206 6.2789e–06 4.0693 1/8 3.3499e–07 4.2210 3.0961e–07 3.9907 3.9130e–07 4.0042 1/16 1.9745e–08 4.0846 1.9323e–08 4.0021 2.4403e–08 4.0031 1/32 1.1750e–09 4.0707 1.1685e–09 4.0475 1.4745e–09 4.0488  1/2 1/2 1.1929e–04  6.8871e–05  9.7399e–05 1/4 5.7849e–06 4.3660 4.5533e–06 3.9189 5.7675e–06 4.0779 1/8 3.1036e–07 4.2203 2.8654e–07 3.9901 3.5923e–07 4.0050 1/16 1.8246e–08 4.0883 1.7849e–08 4.0048 2.2357e–08 4.0061 1/32 1.0507e–09 4.1182 1.0447e–09 4.0946 1.3056e–09 4.0980  3/4 1/2 1.0782e–04  6.2247e–05  8.8031e–05 1/4 5.2457e–06 4.3613 4.1220e–06 3.9166 5.1648e–06 4.0912 1/8 2.8166e–07 4.2191 2.5959e–07 3.9891 3.2142e–07 4.0062 1/16 1.6515e–08 4.0921 1.6146e–08 4.0070 2.0016e–08 4.0053 1/32 9.2416e–10 4.1595 9.1874e–10 4.1354 1.1316e–09 4.1447


## 5. Concluding Remarks

In this paper, a high-order compact finite difference method for a class of time-fractional Black-Scholes equations is presented and analysed. We apply theL2-1σ approximation formula to the Caputo derivative; then we construct a fourth-order compact finite difference approximation for the spatial derivative. We have analysed the solvability, stability, and convergence of the constructed scheme and provided the optimal error estimates. The constructed scheme has the second-order temporal accuracy and the fourth-order spatial accuracy, which improves the temporal accuracy of the method given in [8].



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