A Weighted Average Finite Difference Scheme for the Numerical Solution of Stochastic Parabolic Partial Differential Equations

Dumitru Baleanu,Mehran Namjoo,Ali Mohebbian and Amin Jajarmi

1Department of Mathematics,Faculty of Arts and Sciences,?ankaya University,Ankara,06530,Turkey

2Institute of Space Sciences,Magurele-Bucharest,R 76900,Romania

3Department of Medical Research,China Medical University Hospital,China Medical University,Taichung,40402,Taiwan

4Department of Mathematics,Vali-e-Asr University of Rafsanjan,Rafsanjan,77188-97111,Iran

5Department of Electrical Engineering,University of Bojnord,Bojnord,94531-1339,Iran

ABSTRACT In the present paper,the numerical solution of It? type stochastic parabolic equation with a time white noise process is imparted based on a stochastic finite difference scheme.At the beginning,an implicit stochastic finite difference scheme is presented for this equation.Some mathematical analyses of the scheme are then discussed.Lastly,to ascertain the efficacy and accuracy of the suggested technique,the numerical results are discussed and compared with the exact solution.

KEYWORDS It? equation;stochastic process;finite difference scheme;stability and convergence;consistency

1 Introduction

Stochastic partial differential equations (SPDEs) driven by white noise are one of the essential classes of partial differential equations (PDEs).This class of equations arises in many branches of applied sciences and engineering,such as nonlinear filtering [1],turbulent flows [2],population biology[3],microscopic particle dynamics[4],groundwater flow[5],etc.Few numbers of SPDEs can be solved by analytical techniques [6],most of which cannot be analyzed by well-known analytical schemes suitably.Due to this reason,various numerical methods have been discussed to solve such equations[7–9].For instance,the authors in[10]proposed an explicit scheme to obtain the approximate solution of stochastic equations.In[11],a compact finite difference method for solving a stochastic advection-diffusion equation was proposed.In[12],two techniques on the basis of Saul’yev method and finite difference scheme were suggested for solving linear SPDEs.In [13],explicit and implicit finite difference methods were proposed to obtain the solution of general SPDEs.In[14],a stochastic compact finite difference scheme was suggested for solving a stochastic fractional advection-diffusion equation.In [15],high-resolution finite volume methods were used to solve SPDEs.In [16],the authors proposed a spectral collocation method for the numerical solution of SPDEs driven by infinite dimensional fractional Brownian motions.More than these,some authors used spectral methods for the discretization of spatial variables and applied a Crank-Nicolson scheme or a stochastic Runge-Kutta method for solving the resultant system of stochastic differential equations[17].

In[18,19],the authors investigated the convergence and stability of two stochastic finite difference schemes for a class of SPDEs.In more detail,the study[19]employed a Crank-Nicolson technique for the approximation of second-order derivatives.Although the reported results in[18,19]are interesting in some senses,the solution methods presented are only conditionally stable.To overcome this issue,here we extend a type of finite difference scheme to a stochastic version in order to approximate the solution of a stochastic advection-diffusion equation.To do so,instead of the Crank-Nicolson method used in[19],we consider a convex combination of discretized second-order derivatives in two consecutive time grid points.As a result,the proposed method in our case is unconditionally stable under a necessary condition,so there will be no limitation for the selection of space and time step sizes.This important feature makes the computational cost of our suggested technique less than the other methods available in the literature[18,19].In the following,the main contributions of our study are summarized and highlighted as below:

? In this paper,a stochastic finite difference scheme is developed for the numerical solution of It? type stochastic parabolic equation.

? As a theoretical investigation,some mathematical results for the proposed scheme are studied.

? In addition,the convergence of the suggested technique is discussed,and the necessary conditions for its conditional and unconditional stability are explored.

? Finally,the efficiency of the proposed method is shown by some numerical examples,and its key qualifications are examined as well.

The rest of this paper is structured as follows.An implicit finite difference scheme is proposed in Section 2,where some mathematical analyses are also investigated.Next,some numerical results are given in Section 3.Finally,the paper is closed by some concluding remarks in the last section.

2 Proposed Scheme

In this work,the following problem for the stochastic equation of It? type is considered:

forx∈(0,1)andt∈(0,1].In this problem,the coefficientsρa(bǔ)ndσare constants,andξ(t)indicates a standard Wiener process.Also,the noise term ˙ξ(t) is introduced to present a time white noise.Formally,˙ξ(t)is a Gaussian distribution with zero mean value[20].

Finite difference schemes are the most natural way of solving PDEs numerically.Furthermore,these methods are widely used in approximating the solution of SPDEs like(1).The idea behind these schemes is to discretize the continuous time and space into a finite number of discrete grid points.Then the values of state variables are calculated at any point of the grid.By considering a uniform space gridΔxand time gridΔtin the time-space lattice,the solution of the equation can be estimated at the lattice points.The value of the approximate solution at the point(hΔx,mΔt)is indicated by the random variablevmh,wheremandhare integer.The later stage is to approximate the problem(1)on the mentioned grid.For this purpose,the time and the space derivatives in the SPDE(1)are replaced by the following finite difference approximations:

where 0≤λ≤1.Indeed,we use a forward finite difference scheme for the approximation ofvt(hΔx,mΔt)as

and employ a convex combination of second-order derivatives in the time stepsmandm+1 for the approximation ofvxx(hΔx,mΔt)by

For more details,the interested reader can refer to[21].Substituting the approximations from(2)into(1),we can find

wherer=andΔξm=ξ((m+1)Δt)?ξ(mΔt)is a Gaussian distribution with zero mean value and varianceΔt,i.e.,Δξm～N(0,Δt).

Remark 2.1.In the proposed scheme,the Wiener process increments are not dependent on the statevmh.

Substantially,the convergence of the stochastic difference scheme to the SPDE solution is very important.To achieve this,consider an SPDE in the form ofLu=F,whereinFis an inhomogeneity andLrepresents the differential operator.Suppose that the random variablevmhbe a solution that is approximated by a stochastic finite difference scheme indicated byLmh.By applying the stochastic scheme to this SPDE,we obtain

whereFhnis the approximation of inhomogeneityF.In favor of accessing the consistency,stability,and convergence results,a norm is needed.Because of this,for the sequencev={...,v?1,v0,v1,...},we define the sup–norm as ‖v‖∞=For additional details concerning the concepts of consistency,stability and convergence,see[10].

Definition 2.1.A stochastic finite difference schemeLmh vmh=Fhmis said to be point-wise consistent in mean square with PDELu=Fat point(x,t),if for any continuously differentiable solution?=?(x,t)of this equation,we have

as(Δx,Δt)→(0,0)and(hΔx,(m+1)Δt)→(x,t).

Theorem 2.1.The numerical scheme(5)is consistent in mean square in the sense of Definition 2.1.

Proof.For the smooth function?(x,t),we have

and

Accordingly,

In as much as?(x,t)is a deterministic function,E|L(?)|mh?Lmh ?|2→0 asm,h→∞.Hence,the numerical scheme(5)is consistent with the SPDE(1).

By the assumption that ?vm+1is the Fourier transform ofvm+1,the Fourier inversion formula results in

where

andηis a real variable.We utilize the Von Neumann method to investigate the stability of the stochastic difference scheme.By substituting Eq.(11) into the stochastic difference equation and using the equality of Fourier transformation,one achieves

will be the necessary and sufficient condition for the stability[10].

Theorem 2.2.For the stochastic advection-diffusion Eq.(1),the stochastic scheme(5)is unconditionally stable forλ≥based on the Fourier transformation analysis,and is conditionally stable forλ≤under the conditionr≤

Proof.Substituting(11)into(5),we get

Then we have

Hence,the stochastic difference scheme amplification factor is

Setχ(Δxη)=,soχ(θ)=.Now,by setting the derivative ofχ(θ)equal to zero,the critical points are obtained asθ=0,±π.Then one notes thatχ(0)=1 and

is equivalent to 4r(1?2λ)≤2,clearly ifλ≥then the inequality(18)is always satisfied,and ifλ<12,then the inequality(18)is satisfied only if

And also

is always satisfied.Hence,we see that ifλ≥the scheme(5)is unconditionally stable,and ifλ

Definition 2.2.The stochastic difference schemeLmh vmh=Fhm,which approximates the SPDELu=F,is convergent in mean square at timetwhen(m+ 1)Δtconverges tot,E‖vm+1?um+1‖2→0 for(m+1)Δt=t,Δx→0,andΔt→0.

Theorem 2.3.The numerical scheme(5)for the Eq.(1)is convergent in mean square with respect to‖·‖∞=withr≤andt=(m+1)Δt.

Proof.The stochastic finite difference scheme is given by

The solutionumh+1is represented by the Taylor’s expansionuxx(x,w) with respect to the space variable as follows:

whereβ1,β2,β3,β4,Δ∈(0,1).Then we have

Letzmh=umh?vmh,so we get

It gives that

wherer=Applying E|·|2to the above equation and using the following inequality:

E|X+Y+Z+R|2≤8E|Z|2+8E|Y|2+2E|R|2+4E|X|2,

we have

and so

By introducing the notationΘ1h=uxxxx((h+β1)Δx,s)<∞,Θ2h=uxxxx((h+β2)Δx,s)<∞,Θ3h=uxxxx((h+β3)Δx,s+Δt)<∞,Θ4h=uxxxx((h+β4)Δx,s+Δt)<∞,Θ5h=uxxxx(hΔx,s+δΔt)<∞,ψ1h=uxx(x,s)<∞,taking into account

and the usage of suppositionr≤one concludes that

Therefore,

and

It gives that

WhenΔt→0,we have

Here,it is worth mentioning that according to the inequality(32),the error of the proposed scheme(5)is of first order with respect to the time.

3 Numerical Results and Discussion

In this part,we demonstrate the efficacy and accuracy of the suggested technique,developed in the previous section,by solving some numerical examples.Indeed,we investigate the theoretical consequences of previous section about the stability and convergence of the proposed scheme(5).In more detail,we discuss the convergence of the scheme(5)for each example and explore the necessary conditions for its conditional and unconditional stability.Numerical results in this section verify the previously presented theoretical analysis.

Example 3.1.Consider an SPDE in the following form:

supplemented with the initial and boundary conditions

The exact solution is

if there is no noise term.Following the proposed idea developed in this paper,the stochastic finite difference scheme can be written as follows:

wherer=To qualify the numerical results obtained in this example,the exact and numerical solutions are plotted in Fig.1.LetMandNbe the total numbers of grid points for the space and time discretization,respectively.If we setρ=0.01,σ=1,andM=125(Δx=0.008),then according to Theorem 2.2,the stochastic finite difference scheme (36) is unconditionally stable for allλ≥(see Table 1),and it is conditionally stable forλ

Figure 1:Comparison between the exact solution and the stochastic numerical solution of(33)with ρ=0.01,Δx=0.008,σ=1,Δt=0.008,λ=0.75(right figure),and σ=1.5,Δx=0.01,ρ=0.001,Δt=0.01,λ=0:25(left figure)(Example 3.1)

Table 1: Examination of unconditional stability for the stochastic scheme(36)(Example 3.1)

Table 2: Examination of conditional stability for the stochastic scheme(36)(Example 3.1)

Table 3: Absolute errors of the numerical scheme (36) for Example 3.1 with σ=1.5,Δx=0.01,ρ=0.001,Δt=0.01,0.04,0.05,and λ=0.25

Example 3.2.Let us consider the following problem for the next example:

with the exact solution

In Fig.2,the exact solution and the stochastic numerical solution of (37) are compared for the two sets ofM=100,N=100,λ=0.5 (left plot) andM=120,N=120,λ=0.55 (right plot).More comparisons between the exact and the numerical solutions are given in Fig.3 for the values ofρ=1,σ=1,λ=0.55,Δx=,Δt=0.01 (left plot) andρ=1,σ=1,λ=0.5,Δx=0.01,Δt=0.02(right plot).From the numerical results in Figs.2 and 3,one can see the high accuracy of the presented method for solving the SPDE (37).In Table 4,the unconditional stability of the proposed scheme(5)is shown forλ≥.Forλ=0.4 andM=100,Table 5 portrays the conditional stability of the suggested technique(5)whenN≥4000,a fact which is also shown in Fig.4.In addition,the absolute errors of the numerical scheme(5)withM=100,N=100,λ=0.5,Δx=0.01,andΔt=0.01,0.02,0.2 are reported in Table 6.

Figure 2:Comparison between the exact solution and the stochastic numerical solution of(37)with M=100,N=100,λ=0.5(left figure),and M=120,N=120,λ=0.55(right figure)(Example 3.2)

Figure 3:Comparison between the exact solution and the stochastic numerical solution of(37)with ρ=1,σ=1,λ=0.55,Δx=,Δt=0.01 (left plot),and ρ=1,σ=1,λ=0.5,Δx=0.01,Δt=0.02(rightplot)(Example 3.2)

Table 4: Examination of unconditional stability for the stochastic scheme(5)(Example 3.2)

Table 5: Examination of conditional stability for the stochastic scheme(5)(Example 3.2)

Figure 4: (Continued)

Figure 4:Display of conditional stability for various values of N=400,1000,4000,4100(Example 3.2)

Table 6:Absolute errors of the numerical scheme(5)for Example 3.2 with M=100,N=100,λ=0.5,Δx=0.01,and Δt=0.01,0.02,0.2

Example 3.3.As the third example,consider the following problem:

Fig.5 shows the approximation of SPDE(39)using the stochastic difference scheme(5)with the valuesN=50,60,80,100.In Table 7,the unconditional stability of the proposed method is depicted forλ≥To test the conditional stability forρ=0.01,M=100,andλ=0.3,simulation results of (39) for various values ofNare presented in Fig.5 and Table 8.According to these results for Example 3.3,it is apparent that the stochastic finite difference scheme(5) is stable whenN≥80,a fact which coincides with the stability condition provided by Theorem 2.2.

Figure 5:Display of conditional stability for various values of N=50,60,80,100(Example 3.3)

Table 7: Examination of unconditional stability for the stochastic scheme(5)(Example 3.3)

Table 8: Examination of conditional stability for the stochastic scheme(5)(Example 3.3)

4 Conclusion

This study presented a numerical method based on the weighted average finite difference scheme for the solution of SPDEs.In this paper,we provided some mathematical analyses for the proposed numerical scheme.To ascertain the accuracy and efficacy of the proffered technique,we presented three numerical examples with different boundary conditions,and compared the associated numerical results with the exact solution.Additionally,we explored the necessary conditions for the conditional and unconditional stability of the presented method and verified the theoretical consequences in this regard by some figures and tables.

Future works can be focused on applying some new discrete schemes,such as those discussed in [22] with a second-order time convergence rate,for the numerical solution of stochastic problem studied in this paper.

Acknowledgement: The authors would like to express their deep gratitude to Dr.Fahimeh Akhavan Ghassabzade for her valuable assistance during the development of this research work.

Funding Statement:The authors received no specific funding for this study.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.