Finite Volume Method I.

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Technical Mathematics

Synopsis:

The subject deals with the basics of the finite volume method for discretization of partial differential equations (PDE) with special attention to the equations arising from the fluid mechanics.

1.Conservation laws in the fluid mechanics, the formulation with PDEs. Linear convection, initial and initial-boundary value problems (IBVP), analytical solution.

2.Non-linear Burgers’ equation, formation of shocks, strong and weak solutions.

3.Finite volume method (FVM) for one-dimensional problems, basic low order schemes, stability, consistency, and convergence (Lax theorem) for linear problems.

4.Higher order schemes, concept of artificial viscosity.

5.Stability and convergence for non-linear scalar problems. monotone schemes, TVD schemes, reconstructions with limiters.

6.Linear hyperbolic systems, characteristic variables, formulation of IBVP, boundary conditions, basic numerical methods for hyperbolic systems.

7.Non-linear hyperbolic system, Riemann problem, Godunov’s method, numerical flux based on the (approximate) Riemann solvers.

8.Shallow water equation, HLL flux, application of FVM to the solution shallow water equations.

9.Euler equation, numerical fluxes for Euler equations (Roe, HLL, AUSM)

10.Approximation of source and diffusive terms.

11.Implicit method for flows of compressible fluids.

12.Formulation and demonstration of solution of selected problems.

13.Formulation and demonstration of solution of selected problems.

Requirements:

Syllabus of lectures:

1.Conservation laws in the fluid mechanics, the formulation with PDEs. Linear convection, initial and initial-boundary value problems (IBVP), analytical solution.

2.Non-linear Burgers’ equation, formation of shocks, strong and weak solutions.

3.Finite volume method (FVM) for one-dimensional problems, basic low order schemes, stability, consistency, and convergence (Lax theorem) for linear problems.

4.Higher order schemes, concept of artificial viscosity.

5.Stability and convergence for non-linear scalar problems. monotone schemes, TVD schemes, reconstructions with limiters.

6.Linear hyperbolic systems, characteristic variables, formulation of IBVP, boundary conditions, basic numerical methods for hyperbolic systems.

7.Non-linear hyperbolic system, Riemann problem, Godunov’s method, numerical flux based on the (approximate) Riemann solvers.

8.Shallow water equation, HLL flux, application of FVM to the solution shallow water equations.

9.Euler equation, numerical fluxes for Euler equations (Roe, HLL, AUSM)

10.Approximation of source and diffusive terms.

11.Implicit method for flows of compressible fluids.

12.Formulation and demonstration of solution of selected problems.

13.Formulation and demonstration of solution of selected problems.

Syllabus of tutorials:

Study Objective:

Study materials:

J. Fürst, K. Kozel: Numerické metody řešení problémů proudění I, skripta ČVUT, 2001

J. Fořt, K. Kozel: Numerické metody řešení problémů proudění II, skripta ČVUT, 2002

J. Fořt, K. Kozel, P. Louda, J. Fürst: Numerické metody řešení problémů proudění III, skripta ČVUT, 2004

J.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics, Springer 2002

J. Blazek: Computational Fluid Dynamics: Principles and Applications, Elsevier, 2001

E.F.Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer 2009

R. J. LeVeque: Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, New York, 2007

C. Hirsh: Numerical Computation of Internal & External Flows, vol. 1, Elsevier, 2007

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: