# Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

Authors: Qiyong Li; Siqing Gan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is concerned with the stability of analytical and numerical solutions fornonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize Δt=τ/m when 1/2≤θ≤1, and they are exponentially mean-square stable if the stepsize Δt∈(0,Δt0) when 0≤θ<1. Finally, some numerical experiments are given to illustrate the theoretical results.


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## Body

## 1. Introduction

Models that incorporate jumps have become increasingly popular in finance and several areas of science and engineering. In particular, they are used in mathematical finance in order to simulate asset prices, interest rates, and volatilities [1, 2]. Jump models also arise in many other application areas and have proved successful at describing unexpected, abrupt changes of state [3]. So, it is valuable to investigate the properties of the solutions of these problems.As is well known, explicit solutions of stochastic differential equations can rarely be obtained. It is necessary to construct efficient numerical methods to solve these equations. In recent years, many researchers worked on the construction of numerical schemes for stochastic ordinary differential equations (SODEs) (see [4, 5], and their references) and stochastic delay differential equations (SDDEs), see, for example, [6–11] and references therein. For SODEs with jumps, the strong convergence and mean-square stability of some semi-implicit numerical methods are investigated in [12–15]. A compensated split-step backward Euler method for SODEs with jumps is introduced in [12] and proved to satisfy a better stability property than the split-step backward Euler method.For SDDEs with jumps, most of the existing work is concerned about convergence property of numerical methods, see, for example, [16–19]. There are few results on stability property, which motivates our work. In [20], Tan and Wang investigated the mean-square stability of the explicit Euler method for linear SDDEs with jumps. The aim of our paper is to investigate the mean-square stability of the compensated stochastic θ methods for nonlinear SDDEs with jumps.This paper is organized as follows. In Section2, we obtain a stability result for the analytical solution of (2.1). In Section 3, the compensated stochastic θ methods are constructed to solve problem (2.1). In Section 4, our main results will be stated and proved. It is shown that the compensated stochastic θ methods inherit mean-square stability of the exact solution. More precisely, the methods are mean-square stable for any stepsize Δt=τ/m when 1/2≤θ≤1, and they are exponentially mean-square stable if the stepsize Δt∈(0,Δt0) when 0≤θ<1. Moreover, when θ=1, the method is exponentially mean-square stable for every stepsize Δt=τ/m. Finally, some numerical experiments are reported to illustrate the theoretical results.


## 2. Stability of the Analytical Solution

Throughout this paper, we let𝒞([−τ,0]; Rd) denote the family of continuous functions from [-τ,0] to Rd equipped with the norm ∥φ∥:=sup-τ≤s≤0|φ(s)|, where |·| is the Euclidean norm in ℝd. Denoted by 𝒞ℱ0b([-τ,0];ℝd) the family of all bounded, ℱ0 measurable, 𝒞([-τ,0]; Rd) valued stochastic variables. The inner product of x,y in ℝd is denoted by 〈x,y〉. If A is a vector or matrix, its transpose is denoted by AT. If A is a matrix, its trace norm is denoted by |A|=trace(ATA).We consider the nonlinear SDDEs with jumps in Itô’s sense of the form:(2.1)dx(t)=f(x(t),x(t-τ))dt+g(x(t),x(t-τ))dW(t)+h(x(t-),x(t--τ))dN(t),t>0,
with initial data x(t)=φ(t),t∈[-τ,0] and τ>0 is a constant, x(t-) denotes lims→t-x(s), f:ℝd×ℝd→ℝd, g:ℝd×ℝd→ℝd×m, h:ℝd×ℝd→ℝd are continuous functions, and φ(t)∈𝒞ℱ0b([-τ,0];ℝd). W(t) is an m-dimensional Wiener process defined on the complete probability space (Ω,ℱ,{ℱt}t≥0,P) with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ0 contains all P-null sets). N(t) is a scalar Poisson process with parameter λ defined on the same probability space. Assume that W(t) and N(t) are independent of ℱ0. Moreover, we assume that φ(t) is ℱ0-measurable and right continuous with 𝔼∥φ∥2<∞. We also assume that f(0,0)=0,g(0,0)=0 and h(0,0)=0, so problem (2.1) admits a zero solution x(t)≡0.Definition 2.1 (see [21]).
The zero solution of (2.1) is said to be pth moment exponentially stable if there is a pair of positive constants λ and C such that
(2.2)E|x(t)|p≤C‖φ‖exp⁡-λ(t-t0),t≥t0,
for all φ(t)∈𝒞ℱ0b([-τ,0];ℝd). When p=2, it is usually said to be exponentially mean-square stable.Now, we establish a mean-square stability condition for problem (2.1).Theorem 2.2.
Suppose that there are some constantsαi,βi,γi,i=1,2, such that
(2.3)〈x1-x2,f(x1,y)-f(x2,y)〉≤α1|x1-x2|2,(2.4)|f(x,y1)-f(x,y2)|≤α2|y1-y2|,(2.5)|g(x1,y1)-g(x2,y2)|2≤β1|x1-x2|2+β2|y1-y2|2,(2.6)|h(x1,y1)-h(x2,y2)|2≤γ1|x1-x2|2+γ2|y1-y2|2,
for all x1,y1,x2,y2∈ℝd. If
(2.7)α=2α1+2α2+β1+β2+λ(1+2γ1+2γ2)<0,
then the zero solution of (2.1) is exponentially mean-square stable.Proof.
Lett≥0, δ>0, it follows from Itô’s formula that
(2.8)|x(t+δ)|2=|x(t)|2+∫tt+δ(2〈x(s),f(x(s),x(s-τ))〉+|g(x(s),x(s-τ))|2)ds+∫tt+δ2〈x(s),g(x(s),x(s-τ))〉dW(s)+∫tt+δ(2〈x(s-),h(x(s-),x(s--τ))〉+|h(x(s-),x(s--τ))|2)dÑ(s)+λ∫tt+δ(2〈x(s),h(x(s),x(s-τ))〉+|h(x(s),x(s-τ))|2)ds,
where Ñ(t)=N(t)-λt. Taking expectation and using the properties of Itô integral give
(2.9)E|x(t+δ)|2=E|x(t)|2+E∫tt+δ(2〈x(s),f(x(s),x(s-τ))〉+|g(x(s),x(s-τ))|2)ds+λE∫tt+δ(2〈x(s),h(x(s),x(s-τ))〉+|h(x(s),x(s-τ))|2)ds.
From (2.3) and (2.4), we have
(2.10)2E〈x(s),f(x(s),x(s-τ))〉=2E〈x(s),f(x(s),x(s-τ))-f(0,x(s-τ))〉+2E〈x(s),f(0,x(s-τ))〉≤2α1E|x(s)|2+2α2E|x(s)∥x(s-τ)|≤2α1E|x(s)|2+2α2sup⁡s-τ≤u≤sE|x(u)|2.
Similarly, (2.5) and (2.6) yield
(2.11)E|g(x(s),x(s-τ))|2≤(β1+β2)sup⁡s-τ≤u≤sE|x(u)|2,E|h(x(s),x(s-τ))|2≤(γ1+γ2)sup⁡s-τ≤u≤sE|x(u)|2,2E〈x(s),h(x(s),x(s-τ))〉≤(1+γ1+γ2)sup⁡s-τ≤u≤sE|x(u)|2.
Substituting (2.10)-(2.11) into (2.9) yields
(2.12)E|x(t+δ)|2≤E|x(t)|2+∫tt+δ(2α1E|x(s)|2+(2α2+β1+β2+λ(1+2γ1+2γ2))sup⁡s-τ≤u≤sE|x(u)|2)ds.
Let ν(t)=𝔼|x(t)|2, β=2α2+β1+β2+λ(1+2γ1+2γ2), we have
(2.13)D+ν(t)≤2α1ν(t)+βsup⁡t-τ≤u≤tν(u),
where
(2.14)D+ν(t)=limsup⁡δ↘0ν(t+δ)-ν(t)δ.
Moreover, α<0 implies -2α1>β≥0. By Lemma  1.1 in [22], there exist positive constants v and k such that
(2.15)ν(t)≤ke-vt,t≥0.
Hence, the theorem is proven.Based on the above result, we are going to study the stability of numerical methods for (2.1) in the following sections.


## 3. Compensated Stochasticθ Methods for Nonlinear SDDEs with Jumps

Since the compensated Poisson processÑ(t)=N(t)-λt is a martingale satisfying the property(3.1)E(Ñ(t+s)-Ñ(t))=0,E|Ñ(t+s)-Ñ(t)|2=λs,t,s≥0,
we rewrite problem (2.1) in an equivalent form:(3.2)dx(t)=f̃(x(t),x(t-τ))dt+g(x(t),x(t-τ))dW(t)+h(x(t-),x(t--τ))dÑ(t),t>0,
where f̃(x,y) is defined as(3.3)f̃(x,y):=f(x,y)+λh(x,y).
Applying the stochastic θ methods to (3.2) leads to the following compensated stochastic θ methods:(3.4)Xn+1=Xn+(1-θ)f̃(Xn,Xn-m)Δt+θf̃(Xn+1,Xn-m+1)Δt+g(Xn,Xn-m)ΔWn+h(Xn,Xn-m)ΔÑn.
Here, Xn denotes that the approximation to x(tn), θ is a parameter with 0≤θ≤1, Δt:=tn+1-tn is the stepsize which satisfies τ=mΔt for a positive integer m, ΔWn:=W(tn+1)-W(tn) and ΔÑn:=Ñ(tn+1)-Ñ(tn). In particular,(3.5)Xl=φ(lΔt),l≤0.
Note that for θ>0, the numerical solutions in (3.4) are defined by implicit equations. However, due to the one-sided Lipschitz condition (2.3), (3.4) has a unique solution, with probability one, for all θΔt(α1+λγ1)<1, see, for example, [23, Theorem  14.2] and (19) in [12].Remark 3.1.
Sinceα<0 implies α1+λγ1<0, then the compensated stochastic θ methods (3.4) produce a well-defined, unique solution if the stability condition α<0 holds.Definition 3.2.
For a give stepsizeΔt=τ/m, a numerical method on the nonlinear SDDEs with jumps (2.1) is said to be exponentially mean-square stable, if there exist positive constants N and γ, such that the numerical solution Xn produced by this method satisfies
(3.6)E|Xn|2≤N‖φ‖e-γ(tn-t0),
for all initial data φ∈𝒞ℱ0b([-τ,0];ℝd).Definition 3.3.
For a give stepsizeΔt=τ/m, a numerical method on the nonlinear SDDEs with jumps (2.1) is said to be mean-square stable if the numerical solution Xn produced by this method satisfies
(3.7)lim⁡n→∞E|Xn|2=0.


## 4. Stability Analysis of the Numerical Solutions

In this section, we study mean-square stability and exponentially mean-square stability of the compensated stochasticθ methods (3.4). Now, we present the main results of the paper.Theorem 4.1.
Suppose that (2.3)–(2.7) hold. If 1/2≤θ≤1, then the compensated stochastic θ methods are mean-square stable for every stepsize Δt=τ/m.Proof.
It follows from (3.4) that
(4.1)|Xn+1-θΔtf̃(Xn+1,Xn-m+1)|2=|Xn-θΔtf̃(Xn,Xn-m)|2+2Δt〈Xn,f̃(Xn,Xn-m)〉+(Δt)2(1-2θ)|f̃(Xn,Xn-m)|2+|g(Xn,Xn-m)ΔWn|2+|h(Xn,Xn-m)ΔÑn|2+Mn,
where
(4.2)Mn=2〈Xn+(1-θ)Δtf̃(Xn,Xn-m),g(Xn,Xn-m)ΔWn〉+2〈Xn+(1-θ)Δtf̃(Xn,Xn-m),h(Xn,Xn-m)ΔÑn〉+2〈g(Xn,Xn-m)ΔWn,h(Xn,Xn-m)ΔÑn〉.
Thus, for 1/2≤θ≤1, we have
(4.3)|Xn+1-θΔtf̃(Xn+1,Xn-m+1)|2≤|Xn-θΔtf̃(Xn,Xn-m)|2+2Δt〈Xn,f̃(Xn,Xn-m)〉+|g(Xn,Xn-m)ΔWn|2+|h(Xn,Xn-m)ΔÑn|2+Mn.
It follows from (2.3), (2.4), and (2.6) that
(4.4)2〈Xn,f̃(Xn,Xn-m)〉=2〈Xn,f(Xn,Xn-m)〉+2λ〈Xn,h(Xn,Xn-m)〉≤2α1|Xn|2+α2|Xn|2+α2|Xn-m|2+λ(|Xn|2+γ1|Xn|2+γ2|Xn-m|2).
Note that 𝔼(ΔWn)=0, 𝔼(ΔÑn)=0 and 𝔼(ΔÑn)2=λΔt. Furthermore, Xn and Xn-m are all ℱtn-measurable. Therefore, we can easily obtain
(4.5)E|g(Xn,Xn-m)ΔWn|2=ΔtE|g(Xn,Xn-m)|2,(4.6)E|h(Xn,Xn-m)ΔÑn|2=λΔtE|h(Xn,Xn-m)|2,(4.7)EMn=0.
Taking expectation on both sides of (4.3) and substituting (4.4)–(4.7) into (4.3), we have
(4.8)E|Xn+1-θΔtf̃(Xn+1,Xn-m+1)|2≤E|Xn-θΔtf̃(Xn,Xn-m)|2+Δt(2α1+α2+β1+λ+2λγ1)E|Xn|2+Δt(α2+β2+2λγ2)E|Xn-m|2.
Consequently, by the recursion of inequality (4.8), we have
(4.9)E|Xn+1-θΔtf̃(Xn+1,Xn-m+1)|2≤E|Xn-θΔtf̃(Xn,Xn-m)|2+Δt(2α1+α2+β1+λ+2λγ1)E|Xn|2+Δt(α2+β2+2λγ2)E|Xn-m|2≤E|Xn-1-θΔtf̃(Xn-1,Xn-m-1)|2+Δt(2α1+α2+β1+λ+2λγ1)∑j=n-1nE|Xj|2+Δt(α2+β2+2λγ2)∑j=n-1nE|Xj-m|2≤⋯≤E|X0-θΔtf̃(X0,X-m)|2+Δt(2α1+α2+β1+λ+2λγ1)∑j=0nE|Xj|2+Δt(α2+β2+2λγ2)∑j=0nE|Xj-m|2.
Noting that ∑j=0n𝔼|Xj-m|2=∑j=-m-1𝔼|Xj|2+∑j=0n-m𝔼|Xj|2 and τ=mΔt, we derive from (4.9) that
(4.10)E|Xn+1-θΔtf̃(Xn+1,Xn-m+1)|2≤E|X0-θΔtf̃(X0,X-m)|2+Δt(2α1+2α2+β1+β2+λ(1+2γ1+2γ2))∑j=0nE|Xj|2+τ(α2+β2+2λγ2)×max⁡-m≤j≤0E|Xj|2.
Rearranging (4.10) and using the notation α in (2.7), we obtain
(4.11)E|Xn+1-θΔtf̃(Xn+1,Xn-m+1)|2-αΔt∑j=0nE|Xj|2≤E|X0-θΔtf̃(X0,X-m)|2+(α2+β2+2λγ2)τmax⁡-m≤j≤0E|Xj|2.Since 𝔼∥φ∥2<∞ and α<0, we then derive that the series ∑j=0∞𝔼|Xj|2 is convergent, which implies limn→∞𝔼|Xn|2=0. Consequently, for 1/2≤θ≤1, the compensated stochastic θ methods are mean-square stable for any stepsize Δt=τ/m.In order to investigate the exponential stability of the numerical methods, we need the following lemma which is Theorem  1 in [24].Lemma 4.2 (see [24]).
Suppose, for some fixed integerN≥0, that tn=t0+nΔt for some Δt>0 and {vn}-N∞ is a sequence of positive numbers that satisfies
(4.12)vn+1-vnΔt≤-αΔtvn+βΔtmax⁡j∈Jvn+jforn∈N
with N=0 if βΔt=0, where 𝒥:={-N,…,-1,0}. If
(4.13)0≤βΔt<αΔt,0<αΔtΔt<1,
then vn≤{maxj∈𝒥vj}exp{-v+(tn-t0)}, where v+>0 is a constant.Now, we present the result as follows.Theorem 4.3.
Suppose that (2.3)–(2.7) hold and the drift coefficient f satisfies the linear growth condition, that is, there is a constant D such that
(4.14)|f(x,y)|2≤D(|x|2+|y|2).
Define Δt1=-α/2(1-θ)2(2D+λ2(γ1+γ2)),Δt2=-(2α1+α2+β1+λ+2λγ1)/2(1-θ)2(D+λ2γ1) and Δt3=inf{Δt>0:P(θ,Δt)<0}, where P(θ,Δt)=2(1-θ)2(D+λ2γ1)(Δt)2+((1-θ)(2α1+α2+λ+λγ1)+β1+λγ1)Δt+1. If 0≤θ<1, and the stepsize Δt∈(0,Δt0) with Δt0=min{Δt1,Δt2,Δt3}, then the compensated stochastic θ methods are exponentially mean-square stable.Proof.
We derive from (3.4) that
(4.15)|Xn+1-θΔtf̃(Xn+1,Xn-m+1)|2=|Xn+(1-θ)Δtf̃(Xn,Xn-m)+g(Xn,Xn-m)ΔWn+h(Xn,Xn-m)ΔÑn|2.
Hence, we have
(4.16)|Xn+1|2≤|Xn|2+(1-θ)2(Δt)2|f̃(Xn,Xn-m)|2+|g(Xn,Xn-m)ΔWn|2+|h(Xn,Xn-m)ΔÑn|2+2θΔt〈Xn+1,f̃(Xn+1,Xn-m+1)〉+2(1-θ)Δt〈Xn,f̃(Xn,Xn-m)〉+Mn,
where Mn is defined as (4.2). From (2.6), (3.3), and (4.14), we obtain
(4.17)|f̃(Xn,Xn-m)|2=|f(Xn,Xn-m)+λh(Xn,Xn-m)|2≤2(D(|Xn|2+|Xn-m|2)+λ2(γ1|Xn|2+γ2|Xn-m|2)).
Substituting (4.4)–(4.7) and (4.17) into (4.16), and taking expectation, we have
(4.18)E|Xn+1|2≤E|Xn|2+2(1-θ)2(Δt)2((D+λ2γ1)E|Xn|2+(D+λ2γ2)E|Xn-m|2)+Δt(β1E|Xn|2+β2E|Xn-m|2)+λΔt(γ1E|Xn|2+γ2E|Xn-m|2)+θΔt((2α1+α2+λ(1+γ1))E|Xn+1|2+(α2+λγ2)E|Xn-m+1|2)+(1-θ)Δt((2α1+α2+λ(1+γ1))E|Xn|2+(α2+λγ2)E|Xn-m|2),
which yields
(4.19)(1-θΔt(2α1+α2+λ(1+γ1)))E|Xn+1|2≤(1-θΔt(2α1+α2+λ(1+γ1)))E|Xn|2+(2(1-θ)2(D+λ2γ1)(Δt)2(2α1+α2+β1+λ+2λγ1)Δt+2(1-θ)2(D+λ2γ1)(Δt)2)E|Xn|2+((α2+β2+2λγ2)Δt+2(1-θ)2(D+λ2γ2)(Δt)2)max⁡n-m≤i≤n-m+1E|Xi|2.
Hence,
(4.20)E|Xn+1|2-E|Xn|2Δt≤-AE|Xn|2+Bmax⁡n-m≤i≤n-m+1E|Xi|2,
where
(4.21)A=-2α1+α2+β1+λ+2λγ1+2(1-θ)2(D+λ2γ1)Δt1-θΔt(2α1+α2+λ(1+γ1)),B=α2+β2+2λγ2+2(1-θ)2(D+λ2γ2)Δt1-θΔt(2α1+α2+λ(1+γ1)).
By Lemma 4.2, we derive that the methods are exponentially mean-square stable if
(4.22)0≤B<A,0<AΔt<1.
That is,
(4.23)Δt<-α2(1-θ)2(2D+λ2(γ1+γ2)),Δt<-2α1+α2+β1+λ+2λγ12(1-θ)2(D+λ2γ1),P(θ,Δt)>0,
where P(θ,Δt)=2(1-θ)2(D+λ2γ1)(Δt)2+((1-θ)(2α1+α2+λ+λγ1)+β1+λγ1)Δt+1. Since P(θ,0)=1, there must exist Δt3>0 such that P(θ,Δt)>0 when Δt<Δt3. On the other hand, if P(θ,Δt)≥0 is always true, we then define Δt3 as ∞. Therefore, let Δt1=-α/2(1-θ)2(2D+λ2(γ1+γ2)), Δt2=-(2α1+α2+β1+λ+2λγ1)/2(1-θ)2(D+λ2γ1), Δt3=inf{Δt>0:P(θ,Δt)<0} and Δt0=min{Δt1,Δt2,Δt3}, then (4.22) holds when Δt∈(0,Δt0), which completes the proof of Theorem 4.3.By the proof of Theorem4.3, we can easily obtain the following result.Theorem 4.4.
Suppose that (2.3)–(2.7) hold. If θ=1, then the compensated stochastic θ-method is exponentially mean-square stable for every stepsize Δt=τ/m.


## 5. Numerical Examples

The purpose of this section is to illustrate our theoretical results presented in the previous section by numerical experiments. We first consider the following nonlinear scalar SDDEs with jumps:(5.1)dx(t)=(-4x(t)-x3(t)+x(t-1))dt+sin⁡(x(t-1))dW(t)-x(t-)dN(t),t>0,x(t)=1,t∈[-1,0],
where N(t) is a scalar Poisson process with parameter λ=1. In this case, (2.3)–(2.6) are satisfied with α1=-4,α2=1,β1=0,β2=1,γ1=1,γ2=0, and τ=1. So we have α=-2 in (2.7), which guarantees mean-square stability of the zero solution of (5.1) by Theorem 2.2.The following numerical experiments will show how the parameterθ and the stepsize Δt influence the mean-square stability of the compensated stochastic θ methods. We simulate the expectation of |Xn|2 by using 1000 trajectories, that is,(5.2)E|Xn|2≈11000∑i=11000|Xn(ωi)|2.Theorem4.1 shows that the compensated stochastic θ methods are mean-square stable for every stepsize Δt=τ/m when 1/2≤θ≤1. In Figure 1, we use (3.4) to solve (5.1) and choose the parameter θ with different values 0.5 and 0.8, and we take the stepsize Δt=1/8,1/4,1/2, and 1, respectively. We can find that the compensated stochastic θ methods are mean-square stable with these stepsizes.Fixedθ=0.5 (a) and θ=0.8 (b) with different stepsize Δt=1/8,1/4,1/2,1.
(a)(b)Theorem4.3 shows that the compensated stochastic θ methods are exponentially mean-square stable if the stepsize Δt∈(0,Δt0) when 0≤θ<1. Now, we consider the following nonlinear scalar SDDEs with jumps:(5.3)dx(t)=-3x(t)dt+sin⁡(x(t-1))dW(t)-x(t--1)dN(t),t>0,x(t)=1,t∈[-1,0],
where N(t) is a scalar Poisson process with parameter λ=1. (2.3)–(2.7) and (4.14) are satisfied with α1=-3,α2=0,β1=0,β2=1,γ1=0,γ2=1,τ=1,D=9, and α=-2. Therefore, the zero solution of (5.3) is exponentially mean-square stable. By Theorem 4.3, we calculate Δt1=1/19(1-θ)2, Δt2=5/18(1-θ)2, and P(θ,Δt)=18(1-θ)2(Δt)2-5(1-θ)Δt+1. It is easy to see that P(θ,Δt)>0 for every Δt>0, then we get Δt3=∞. Therefore, we obtain Δt0=min{Δt1,Δt2,Δt3}=1/19(1-θ)2, which implies that the methods applied to (5.3) have less restrictions on the stepsize as the value of θ increases. Now, we use (3.4) to solve (5.3) and choose the parameter θ=0 and θ=0.2, and we take the stepsize Δt=1/20,1/8,1/2, and 1, respectively. By Theorem 4.3, we compute that Δt0=0.0526 when θ=0 and Δt0=0.0822 when θ=0.2. Figure 2 indicates that both methods are exponentially mean-square stable if the stepsize Δt=0.05, which is well selected in Δt∈(0,Δt0).  It also shows that the methods maybe stable when the stepsize is bigger than Δt0, since both methods are stable when stepsize Δt=0.125, but they are not stable for stepsize Δt=1. This indicates that the restriction of the stepsize Δt0 in Theorem 4.3 is not theoretical optimal. From Figure 2, we also can find that the methods behave better stability when the value of θ increases and the stepsize Δt decreases.Fixedθ=0 (a) and θ=0.2 (b) with different stepsize Δt=1/20,1/8,1/2,1.
(a)(b)



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