Unified treatment of physical meaning, mathematical properties, and (Finite Volume) numerical methods for Conservation Laws.

Parallelization for distributed computing via domain decomposition and MPI.

Recent advances in shock-capturing higher order schemes.

Prof. Vasilios ALEXIADES Ayres 213 974-4922 alexiades@utk.edu

8-10 Lab/Homework assignments: 80% , Parallelization Project : 20%

Contact the Office of Disability Services (2227 Dunford Hall, 974-6087) to coordinate reasonable accommodations for documented disabilities.

I. Parallelization for distributed computing - types of parallelism - speedup, strong and weak scaling - domain decomposition - Message Passing Interface (MPI) library - implementation issues and methodsII. The basic conservation law: \( \frac{\partial u}{\partial t} + \nabla \cdot {\vec F} = S \)Note:For parallelization, code must be in Fortran or C/C++ (or possibly Python, using mpi4py, batchOpenMPI).

Model problems: heat conduction, mass diffusion, air pollution, traffic flow, gas dynamics, porous media flow, ... - derivation from first principles, integral and differential forms - conservation of species - advective and diffusive fluxes - continuity equation - conservation of energy - heat conduction - finite volume discretization in 1-D, 2-D, 3-D - explicit / implicit time stepping - consistency-stability-convergence - method of lines / ODE time integrators III. Parabolic Problems Model problem: the above without advection - diffusion ( F = −D∇u ) - parabolic PDEs - boundary conditions - explicit scheme - CFL condition - super-time-stepping acceleration - advection-diffusion ( F = uV - Du_x ) - explicit scheme - CFL condition - effect of small/large Peclet number - parallelization via domain decomposition IV. Basic ODE integrators V. First Order Hyperbolic Conservation Laws Model problem: the above without diffusion/viscosity/dispersion - wave propagation - 1st order PDEs - method of characteristics - linear advection ( F = uV ) - explicit upwind scheme - CFL condition - implementation - nonlinear advection ( F = a(u) ) - Burgers equation - shock formation - monotonicity - flux limiters - slope limiters - Lax-Wendroff - Godunov - TVD - ENO - WENO schemes - Max Principle Satisfying schemes VI. Elliptic Problems (...if there is time...) Model problem: steady-state versions of the above ( electrostatic potential ) - Laplace equation - elliptic PDEs - boundary conditions - finite volume discretization - numerical solution of AU = b (a couple of methods only) - direct methods for banded systems - fast Poisson solvers

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I also appreciate your constant words of encouragement all along the semester. You really encouraged me to work harder."