In this chapter, the aim is to visualize wave dynamics in one dimension as dictated by the Schrödinger equation. The necessary numerical tools are introduced in the first part of the chapter. Via discretization, the wave function is represented as a column vector and the Hamiltonian, which enters into the Schrödinger equation, as a square matrix. It is also seen how different approximations behave as the numerical wave function reaches the numerical boundary – where artefacts appear. This numerical framework is first used to see how a Gaussian wave packet would change its width in time and, eventually, spread out. Two waves interfering is also simulated. And wave packets are sent towards barriers to see how they bounce back or, possibly, tunnel through to the other side. In the last part of the chapter, it is explained how quantum measurements provide eigenvalues as answers – for any observable physical quantity. This, in turn, is related to what is called the collapse of the wave function. It is also discussed how a quantity whose operator commutes with the Hamiltonian is conserved in time. Finally, the concept of stationary solutions is introduced in order to motivate the following chapter.

Overview of discretizing partial differential equations is provided for selected finite difference methods, which is required for construction of discrete tangent linear and adjoint models.

Chapter 15 describes the use of the Bayesian network (BN) methodology for reliability assessment and updating of structural and infrastructure systems. A brief review of the BN as a graphical representation of random variables and an efficient framework for encoding their joint distribution and its updating upon observations is presented. D-separation rules describing the flow of information within the network upon observation of random variables are described and methods are presented for discretizing continuous random variables, thus allowing the use of efficient algorithms applicable to BNs with discrete nodes. Efficient BN models for components, systems, random fields, and seismic hazard are developed. For time- or space-variant problems, the dynamic Bayesian network is introduced. This model is used in conjunction with structural reliability methods (FORM, SORM, simulation) to develop enhanced BNs to solve reliability problems for structures under time-varying loads. Detailed examples are presented, including post-earthquake risk assessment of a spatially distributed infrastructure system and reliability assessment of a deteriorating structure under stochastic loads. The chapter concludes with a discussion of the potential of the BN as a tool for near-real-time risk assessment and decision support for constructed facilities, and the need for further research and development to realize this potential.

The ad-core module in MRST offers an object-oriented framework for rapid prototyping of new reservoir simulators based on automatic differentiation (AD-OO). The framework simplifies the task of changing and extending existing simulation models in MRST or implementing brand new ones. The MRST textbook (Lie, Cambridge University Press, 2019, Chapter 12) presents a model hierarchy for the black-oil equations, discretized by a standard fully implicit method, and describes how to (automatically) select time steps and configure linear and nonlinear solvers. Herein, we present a further modularization of the AD-OO framework that aims to simplify the implementation of more complex flow models and other types of discretizations and solution strategies. To this end, we view the reservoir simulator as a graph of functional relationships and their dependencies and introduce the new concept of so-called state functions to define these functional relationships and compute discrete quantities required for the linearized governing equations. Using the graph perspective, it is relatively simple to not only visualize and understand the data flow of highly complex reservoir simulators, but also replace components of the graph and/or extend the graph with new branches as needed. The result is a versatile family of reservoir simulators that can easily be configured to run different types of multiphase, multicomponent models and at the same time support a number of different spatial and temporal discretizations. The state-function concept also has a built-in compute cache that helps you to systematically eliminate redundant function evaluations. The chapter explains the new concept in detail and exemplifies its use by showcasing implicit, explicit, and adaptive-implicit versions of the same physical processes. We also demonstrate the use of consistent and high-resolution schemes to improve simulation accuracy. Applications to complex flow physics (EOR models, compositional flow, fractured reservoirs) are discussed in other chapters.

In the problems we have considered so far, we have either ignored the actual material consumption/production (sequential environments) or assumed that materials are consumed/produced at fixed proportions (network environments). There are problems, however, where the proportions in which materials are consumed can vary provided that some specifications are satisfied. This problem, which is termed multiperiod blending or simply blending, is fundamentally different from the ones discussed thus far because it leads to nonlinear models. There are two types of blending problems: (1) different streams/inputs are blended before they are processed/converted (process blending); and (2) streams/inputs are blended to produce final products (product blending). In Section 11.1, we introduce some preliminary concepts and a formal problem statement for product blending. In Section 11.2, we present two alternative formulations for product blending, and in Section 11.3, we present two approximate linear reformulations. We close, in Section 11.4, with a discussion of models for process blending. We focus on the equations necessary to account for the key new features of blending problems: (1) the selection of input materials and their blending in variable proportions, and (2) the requirement to satisfy given property specifications.