This unit begins with a discussion of direct solvers. These include Gaussian Elimination and its simplifications for banded systems, viz. the tri-diagonal matrix algorithm, and the penta-diagonal matrix algorithm. Following a discussion of the reasons why direct solution of large linear systems arising out of discretization of partial differential equations is prohibitive, the reader is introduced to iterative solvers. Several iterative solvers are presented. These include the Jacobi method, the Gauss-Seidel method, the Alternating Direction Implicit (ADI) method, the Stone’s Strongly Implicit method, the Method of Steepest Descent, the Conjugate Gradient method, and the Conjugate Gradient Squared method. All of the aforementioned methods are compared and contrasted by solving a benchmark boundary value problem. An overview of Krylov subspace methods is also included. Methods to linearize nonlinear source terms to attain diagonal dominance and stable convergence are also discussed.

In this unit, the stability and convergence characteristics of classical iterative solvers are explored through formal stability analysis. The student is first introduced to the eigenvalues and the condition number of a matrix. Fourier decomposition and the von Neumann stability analysis procedure are presented to compute eigenvalues of the iteration matrix and the spectral radius of convergence. The concept and methods for pre-conditioning of a matrix to enhance convergence rate are outlined and demonstrated through some examples. Finally, the multi-grid method is introduced as a means to selectively damp out large-wavelength error components. Detailed discussion of the Geometric Multi-Grid method is followed by an overview of the Algebraic Multi-Grid method. Multi-grid scheduling algorithms are also presented and demonstrated.

The finite difference method is extended to parabolic and hyperbolic partial differential equations (PDEs). Specifically, this unit addresses the treatment of the time derivative in commonly encountered PDEs in science and engineering. Both explicit (forward Euler) and implicit (backward Euler) time advancement methods are discussed for both the aforementioned types of PDEs. Formal von Neumann stability analysis is conducted to establish stability criteria for each method. Higher-order implicit methods, such as the Crank-Nicolson method and the time-splitting Alternating Direction Implicit (ADI) method are also discussed. The unit ends with a discussion of higher-order explicit methods for solving ordinary differential equations, such as the Runge-Kutta method and the Adams-Bashforth method, and their utility in solving PDEs using the Method of Lines.

In this unit, the finite volume method (FVM) is introduced. The relationship of the method to fundamental conservation laws is elucidated. The fundamental procedure to derive finite volume equations is discussed for both the Cartesian and the cylindrical coordinate systems. The advection-diffusion equation is introduced, and several well-known fluxing schemes for the treatment of the advective flux are presented and demonstrated. These include upwind schemes of various order and the exponential scheme. Several examples are presented to highlight the conservation property of the FVM and its similarities and differences with the finite difference method. Finally, the formulation of the method is extended to generalized curvilinear coordinates (body-fitted mesh) and demonstrated through an example.

The finite volume method is extended to unstructured mesh topology. The Gauss-divergence theorem, which serves as the foundation of the finite volume method, is first ascribed a physical interpretation. Next, it is used to discretize the generalized advection-diffusion equation using the finite volume method on an arbitrary unstructured mesh. The flux at a cell face is split into normal and tangential components and derivation of both components is presented in detail. Processing of the mesh to generate connectivity information is also discussed, followed by a discussion of the procedures for calculating critical geometric quantities, such as surface area, volume, surface normal, and surface tangents. The steps required to develop an unstructured finite volume code from ground up are enlisted. The unit concludes with an example that demonstrates all of the aforementioned concepts.

This unit begins with a brief review of a few important topics that are pertinent to post-processing of data generated by partial differential equation (PDE) solvers. These include interpolation and numerical integration. Interpolation using Lagrange polynomials and splines is discussed. Several methods for numerical integration are also discussed, with a particular emphasis on Gaussian quadrature. Next, the unit delves into solution of nonlinear equations using the generalized Newton’s method and demonstrates how to use the Newton’s method for solution of nonlinear PDEs. The unit concludes with a discussion of the methods that may be used to solve a coupled set of PDEs, as encountered in a variety of practical problems. Fully coupled and segregated solution approaches are compared and contrasted through an example.