Finite difference methods for differential equations, AMCS 252:

The course covers theory and algorithms for the numerical solution of ordinary differential equations (ODEs) and of partial differential equations (PDEs) of parabolic, hyperbolic, and elliptic type. Theoretical concepts include:

  • accuracy
  • zero-stability
  • absolute stability
  • convergence
  • order of accuracy
  • stiffness
  • conservation
  • CFL condition

 Algorithms covered include:

  • finite differences 
  • steady and unsteady discretization in one and two dimensions
  • Newton methods
  • Runge-Kutta methods
  • linear multistep methods
  • multigrid
  • implicit methods for stiff problems
  • centered and upwind methods for wave equations
  • dimensional splitting and operator splitting


Numerical linear algebra, AMCS 251:

Numerical analysis describes the construction and analysis of efficient discrete algorithms to solve continuous problems using large amounts of data. It represents the basis for the field of numerical analysis, and should, therefore, be learned and mastered as early as possible. We cover:

  • fundamentals: computing with matrices, execution time, norms, SVD
  • factorizations: Cholesky decomposition, QR decomposition, LU decomposition
  • least-square: normal equation, orthonormal equation
  • error analysis: error measures, conditioning of a problem, machine precision, stability of an algorithm
  • eigenvalue problem: power iteration, QR, SVD
  • iterative methods: Arnoldi, GMRES, conjugate gradient


Summation-by-parts operators for partial differential equations, AMCS 394 - Contemporary topic in numerical analysis:

The past decade has seen an explosion in popularity of developing methods with the summation-by-parts (SBP) property. This is because the SBP framework offers a simple, yet powerful methodology for the design and analysis of modern algorithms for the solution of PDEs. The SBP concept was originally developed in the finite difference community with the goal of mimicking finite element energy analysis techniques. In recent years, this simple idea has been exponentially generalized enabling a unifying framework for the stability analysis of many spatial discretizations including finite difference, finite volume, flux reconstruction, and continuous/discontinuous Galerkin finite element methods on structured and unstructured polytope meshes for linear and nonlinear conservation laws on conforming and non-conforming grids. The most important consequence of SBP is that it naturally guides the path to stability and robustness as it mimics continuous stability analysis. The SBP concept provides a strong theoretical framework, that is discretization agnostic, for the analysis of existing schemes and the design of flexible high-order numerical approximations that are robust for complex multi-scale applications. The main topics of the course are:

  • introduction of SBP operators through the stability analysis of model problems equations (advection and advection-diffusion equations)
  • construction of collocated Legendre-Gauss-Lobatto SBP operators
  • Hadamard formalism and extension of SBP operators to nonlinear PDEs
  • one element discretizations analysis and weak imposition of boundary conditions through the simultaneous-approximation terms technique
  • multi-element discretizations and analysis and imposition of weak interface coupling
  • discontinuous collocated Galerkin method and entropy stability for the Burgers', compressible Euler and Navier-Stokes equations
  • hp-adaptation linear and nonlinear (entropy) stability analysis of nonconforming interfaces