Exact and numerical solution of stochastic Burgers equations with variable coefficients

Figure 1.  Two realizations of the stochastic process in equation (28)

Figure 2.  Two realizations of the stochastic process in equation (29)

Figure 3.  (a) and (b): Two realizations of the stochastic mesh resulting from solving equation (7) with $ C(t) = t+1 $ and $ E(t) = t+2 $ for $ t\in[0,2] $ with $ \Delta t = 0.0408 $ when $ z\in[-1,1] $ with $ \Delta z = .1 $. (c) and (d): Two realizations of the stochastic mesh resulting from solving equation (9) with $ B(t) = \exp(t) $, $ R(t) = 1 $ and $ E(t) = 1 $ for $ t\in[0,2] $ with $ \Delta t = 0.0408 $ when $ z\in[-1,1] $ with $ \Delta z = .1 $. Notice the uniformity over space since the noise is space uniform

Figure 4.  Stencil of the numerical scheme with the realization of the incremental trajectory $ dX_t $ when it is positive (a) and negative (b). The stencil is shown for $ \Delta t = k $ and $ \Delta x = h $ which are fixed

Figure 5.  Two realizations of the stochastic meshes (a) and (b), and their respective simulated numerical solutions over those two meshes (c) and (d)

Figure 6.  Two realizations of the two processes $ Z_t $ and $ \dot{Z}_t $ that solve equation (31) (a) and (b), the stochastic meshes (c) and (d), and their respective simulated numerical solutions over those two meshes (e) and (f)

Figure 7.  The relative frequency of the times the absolute error of the SFEM is smaller than the absolute error of the stochastic mesh (SM) method for the solution of (25) at each pair $ (t_i,z_j) $ for $ i = 0,\ldots,m $ and $ j = 0,\ldots,n $ for (a) (m, n) = (20, 20) giving $ P_{\text{max}} = .059 $, (b) (m, n) = (20, 30) giving $ P_{\text{max}} = .053 $, (c) (m, n) = (30, 20) giving $ P_{\text{max}} = .077 $, (d) (m, n) = (30, 30) giving $ P_{\text{max}} = .0597 $, (e) (m, n) = (40, 20) giving $ P_{\text{max}} = .139 $, (f) (m, n) = (40, 30) giving $ P_{\text{max}} = .087 $