The equilibrium state of a chemical system composed of an arbitrary number of thermodynamic components is described by a set of nonlinear equations that can be evaluated only using iterative methods. This chapter provides a detailed description of the process by which the Newton–Raphson method can be applied to solve for the equilibrium distribution of chemical mass in a multicomponent system composed of aqueous, solid, and gaseous phases.

Optimization and root finding are closely aligned techniques for determining the extremums and zeros, respectively, of a function.Newton's method is the workhorse of both types of algorithms for nonlinear functions, and the conjugate-gradient and GMRES methods are also covered.Optimization of linear, quadratic, and nonlinear functions are addressed with and without constraints, which may be equality or inequality.In the linear programming case, emphasis is placed on the simplex method.

Unconstrained multivariate gradient-based minimization is introduced by means of search direction-producing methods, focusing on steepest descent and Newton's method.Issues with both methods are discussed, highlighting what happens in the case of locally nonconvex functions, particularly in Newton's method.Linesearch is introduced, effectively rendering multidimensional optimization into a sequence of one-dimensional searches along the ray of the search directions produced.Linesearch criteria are discussed, such as the Armijo first condition, and efficient ways to cut the step size are discussed.

This chapter explains how you can discretize the basic equations for single-phase, compressible flow by use of the discrete differential and averaging operators introduced in Chapter 4. These operators enable you to implement the flow equations in a compact form similar to the continuous mathematical description. By using automatic differentiation, you can automatically linearize and assemble the corresponding linear system without having to explicitly derive and implement expressions for partial derivatives in the Jacobian matrix. The combination of discrete operators and automatic differentiation with a flexible grid structure, a highly vectorized and interactive scripting language, and a powerful graphical environment, is the main reason MRST has proven to be an efficient tool for developing new proof-of-concept codes. To demonstrate this, we first develop a compact solver for compressible flow, and then extend the basic single-phase model to include pressure-dependent viscosity, non-Newton fluid behavior, and temperature effects.