Three-stage Semi-implicit Stochastic Runge-Kutta Methods For Stochastic Differential Equations

In the last few years, K.Burrage, P.M.Burrage and T.H.Tian have given some methods which have better properties for two-stage, four-stage, five-stage Runge-Kutta methods in [1,2,3,4,6]. P.M.Burrage put forward explicit method with optimal principal error coefficients for three-stage explicit Runge-Kutta methods in her thesis for the doctorate ([4]) in 1999, but she never discussed the numerical stability of this class of methods. T.H.Tian and K.Burrage proved that no MS-stability region of two-stage diagonally implicit stochastic Runge-Kutta exists, and they indicate that obtain methods which have better stability properties and numerical behavior by reducing the magnitude of the eigenvalues of the matrices in the stochastic Runge-Kutta methods.In this paper, we discuss three-stage semi-implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Two stochastic Runge-Kutta methods are presented in this paper. Where W2 method is a semi-implicit method with a very large MS-stability region, W1 method is a semi- implicit method with a larger MS-stability region. All two methods are semi-implicit methods with small global error.In this paper, we discuss numerical methods for computing solutions of stochastic differential equations of the Stratonovich type, general equation is given bywhere the term f(y(t)) is the drift coefficient, the term g(y(t)) is the diffusion coefficient, and W(t) is a Wiener process whose increment ΔW(t) = W(t + Δt) — W(t) is a normal random variable N(0, Δt) .Note similar to the deterministic case, that problems whisch are nonautonomous can always be written in autonomous forms by the addition of new variables to the y vector representing these quantities. Hence, we assume equation (4.8) to be autonomous in order to simplity notations.For solving the Stratonovich stochastic differential equations (4.8) with strong order 1.0, a class of three-stage semi-implicit stochastic Runge-Kutta methods is given byYi = yn + h E OiifiYj) + Ji E bii9(Yj)x i = 1, ? ■ ? ,3.i=i j=i3 33=1(4.9)where Ji ~ N(0,h).From the order conditions for method (4.9) with strong order 1.0 we obtain undetermined coefficient and constructing two concrete numerical methods, given by $ f(Y2)) + y yn + hf(Y2) + M2g(Y2) - k 4/(Y2) + f(Y3))and = ynwhose Butcher tableau is given byand4g(Y2) + g(Y3)).0000001 41 401 200010-120141141666Oil66(4.10)(4.11)0000001 41 407 100001043 3510 7014114166Oil666For being convenient methods (4.10) and (4.11) are called methods Wl and W2. Figure 4.2 give the MS-stability regions of methods Wl and W2. Comparing it with the MS-stability regions of Five methods, we can see the method W2 has the larger MS-stability region than the other methods, and the MS-stability region of Wl is also larger relatively. In addition in order to test the accuracy of the methods, we use to two test equations to compare methods Wl and W2 with the other five methods from global error, these two methods have the smaller globle error, as shown in table 4.1 and table 4.2.Two test equations are given bydy = (y*-l)dt + 2(1 -y2)odWW(O)=?), t€[0,l]anddy = (l-y2)odWy(0)=y0, t€[0,l]h2555l 100250400l 800Wl6.19(-6)2.57(-6)1.01(-6)3.84(-7)1.43(-7)5.21(-8)W25.32(-5)2.66(-5)1.33(-5)6.66(-6)3.33(-6)1.67(-6)Heun8.47(-5)4.20(-5)2.08(-5)1.03(-5)5.11(-6)2.54(-6)R22.46(-4)1.58(-4)1.03(-4)6.90(-5)4.67(-5)3.21(-5)RS2.22(-4)1.57(-4)l.ll(-4)7.87(-5)5.57(-5)3.94(-5)Ml1.40(-3)4.86(-4)1.72(-4)6.10(-5)2.16(-5)7.64(-6)PL1.04(-5)4.30(-6)1.68(-6)6.33(-7)2.34(-7)8.52(-8)Table 4.1 Global errors of the seven methods for equation (4.12)hhl 50l 100l 200l 400l 800Wl6.38(-7)2.26(-7)7.89(-8)2.82(-8)9.98(-9)3.53(-9)W26.64(-7)2.35(-7)8.30(-8)2.93(-8)1.04(-8)3.67(-9)Heun7.99(-7)2.82(-7)9.98(-8)3.53(-8)1.25(-8)4.41(-9)R26.39(-7)2.26(-7)7.98(-8)2.82(-8)9.98(-9)3.53(-9)RS1.60(-4)8.00(-5)4.00(-5)2.00(-5)1.00(-5)5.00(-6)Ml5.98(-7)2.11(-7)7.46(-8)2.64(-8)9.33(-9)3.30(-9)PL7.79(-7)2.82(-7)9.98(-8)3.53(-8)1.25(-8)4.41(-9)Table 4.2 Global errors of the seven methods for equation (4.13) Here all the globle error are based on 500 simulated trajectoriesj 500M=5OoS'(4.14)where J denote standard normal radom variable. We take 500 samples for it. \yN -i/fc)(￡jv)| denote the global error when taking the fcth sample, and stapsizes are respectively 23' 3o"> T55' 555' 455' §5o- ^ coputing with MATLAB7.0 the different errors are obtained, for example, here 6.19 x 10~6 denoted by 6.19(-6).Therefore, in this paper considering numerical stability and arrcuracy of three-stage semi-implicit stochastic Runge-Kutta methods Wl and W2 which we constructed have...