Advanced Numerical Methods for Partial Differential Equations

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Typical Scheduling: 
Every even spring

Analysis and implementation of numerical methods for nonlinear partial differential equations including elliptic, hyperbolic, and/or parabolic problems.

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Topic Outline: 
  • Error Analysis and Convergence of Numerical Methods for Elliptic and Parabolic Problems - Norms and function spaces, regularity, stability, implicit and explicit methods, Gronwall arguments, a priori convergence estimates, preservation of dissipativity and invariance under discretization
  • Error Analysis and Convergence of Numerical Methods for Conservation Laws - Existence and uniqueness of solutions, characteristics, stability, upwinding, shock tracking, shock capturing, Euler and Lagrange coordinates, approximation of characteristics, streamline diffusion, regularization and numerical dissipativity, order of convergence and regularity of solutions, compensated compactness
  • Adaptive Error Control - A posteriori estimates, residual errors, stability and propagation of error, mesh refinement and stepsize selection, efficient computation
  • Miscellaneous Topics as Time Permits - Multilevel methods, solving nonlinear systems of equations