Balance equation

In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain in and out of states or set of states. Global balance The global balance equations (also known as full balance equations) are a set of equations that characterize the equilibrium distribution (or any stationary distribution) of a Markov chain, when such a distribution exists. For a continuous time Markov chain with state space \mathcal{S}, transition rate from state i to j given by q_{ij} and equilibrium distribution given by \pi, the global balance equations are given by ::\pi_i = \sum_{j \in S} \pi_j q_{ji}, or equivalently :: \pi_i \sum_{j \in S\setminus {i}} q_{ij} = \sum_{j \in S\setminus {i}} \pi_j q_{ji}. for all i \in S. Here \pi_i q_{ij} represents the probability flux from state i to state j

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An ALE-based numerical technique for modeling sedimentary basin evolution featuring layer deformations and faults

Adelmo Cristiano Innocenza Malossi, Alfio Quarteroni, Andrea Villa

In this paper we present a numerical tool to simulate dynamics of stratified sedimentary basins, i.e. depressions on the Earth’s surface filled by sediments. The basins are usually complicated by crustal deformations and faulting of the sediments. The balance equations, the non-Newtonian rheology of the sediments, and the depth-porosity compaction laws describe here a model of basin evolution. We propose numerical schemes for the basin boundary movement and for the fault tracking. In addition, a time splitting algorithm is employed to reduce the original model into some simpler mathematical problems. The numerical stability and the other features of the developed methodology are shown using simple test cases and some realistic configurations of sedimentary basins.

Elsevier2011

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