n | m | MMAE for SM | MMAE for SFEM |
20 | 20 | 0.032 | 0.047 |
20 | 30 | 0.033 | 0.042 |
20 | 40 | 0.032 | 0.038 |
30 | 20 | 0.030 | 0.050 |
30 | 30 | 0.030 | 0.046 |
30 | 40 | 0.033 | 0.040 |
40 | 20 | 0.035 | 0.052 |
40 | 30 | 0.033 | 0.047 |
40 | 40 | 0.033 | 0.039 |
We will introduce exact and numerical solutions to some stochastic Burgers equations with variable coefficients. The solutions are found using a coupled system of deterministic Burgers equations and stochastic differential equations.
Citation: |
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 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 $
Table 1.
The maximum values of the mean absolute errors over
n | m | MMAE for SM | MMAE for SFEM |
20 | 20 | 0.032 | 0.047 |
20 | 30 | 0.033 | 0.042 |
20 | 40 | 0.032 | 0.038 |
30 | 20 | 0.030 | 0.050 |
30 | 30 | 0.030 | 0.046 |
30 | 40 | 0.033 | 0.040 |
40 | 20 | 0.035 | 0.052 |
40 | 30 | 0.033 | 0.047 |
40 | 40 | 0.033 | 0.039 |
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Two realizations of the stochastic process in equation (28)
Two realizations of the stochastic process in equation (29)
(a) and (b): Two realizations of the stochastic mesh resulting from solving equation (7) with
Stencil of the numerical scheme with the realization of the incremental trajectory
Two realizations of the stochastic meshes (a) and (b), and their respective simulated numerical solutions over those two meshes (c) and (d)
Two realizations of the two processes
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