Analytical Approximation and Numerical Studies of One-dimensional Elliptic Equation with Random Coefficients
In this work, we study a one-dimensional elliptic equation with a random coefficient and derive an explicit analytical solution. We model the random coefficient with a spatially varying random eld, K(x;w) with known covariance function. We derive the relation between the standard deviation of the solution T(x;w) and the correlation length. We observe that, the standard deviation of the solution, T(x; w), initially increases with the correlation length * up to a maximum value, and decreases beyond that max. We observe a scaling law between T and correlation length. We show that, for a small value of coefficient of variation of the random coe*cient, the solution T(x; w) can be approximated with a Gaussian random eld regardless of the underlying probability distribution of K(x; w). This approximation is valid for large value of K, if the correlation length, of input random eld K(x; w) is small. We compare the analytical results with numerical ones obtained from Monte-Carlo method and polynomial chaos based stochastic collocation method. Under aforementioned conditions, we observe a good agreement between the numerical simulations and the analytical results. For a given random coe*cient K(x; w) with known mean and variance we can quickly estimate the variance of the solution at any location for a given correlation length. If the correlation length is not available which is the case in most practical situations, we can still use this analytical solution to estimate the maximum variance of the solution at any location.
- Research Organization:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- AC05-76RL01830
- OSTI ID:
- 1566759
- Report Number(s):
- PNNL-SA-101711
- Journal Information:
- Applied Mathematical Modelling, Vol. 40, Issue 9-10
- Country of Publication:
- United States
- Language:
- English
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