Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location

In this work, we propose a new collocation method for uncertainty quantification in strongly nonlinear problems. Based on polynomial construction, the traditional probabilistic collocation method (PCM) approximates the model output response, which is a function of the random input parameter, from the Eulerian point of view in specific locations. In some cases, especially when the advection dominates, the model response has a strongly nonlinear profile with a discontinuous shock or large gradient. This nonlinearity in the space domain is then translated to nonlinearity in the random parametric domain, which causes nonphysical oscillation and inaccurate estimation using the traditional PCM. To address this issue, a new location‐based transformed probabilistic collocation method (xTPCM) is developed in this study, inspired by the Lagrangian point of view, in which model response is represented by an alternative variable, i.e., the location of a particular response value, which is relatively linear to the random parameter with a smooth profile. The location is then approximated by polynomial construction, from which a sufficient number of location samples are randomly generated and transformed back to obtain the response samples and to estimate the statistical properties. The advantage of the xTPCM is demonstrated through applications to multiphase flow and solute transport in porous media, which shows that the xTPCM achieves higher statistical accuracy than does the PCM, and produces more reasonable realizations without oscillation, while computational effort is greatly reduced compared to the direct sampling Monte Carlo method.