Scientific Computing I

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1 Scientific Computing I Module 10: Case Study Computational Fluid Dynamics Michael Bader Winter 2012/2013 Module 10: Case Study Computational Fluid Dynamics, Winter 2012/2013 1

2 Fluid mechanics as a Discipline Prominent discipline of application for numerical simulations: experimental fluid mechanics: wind tunnel studies, laser Doppler anemometry, hot wire techniques,... theoretical fluid mechanics: investigations concerning the derivation of turbulence models, e.g. computational fluid mechanics (CFD): numerical simulations Many fields of application: aerodynamics: aircraft design, car design,... thermodynamics: heating, cooling,... process engineering: combustion material science: crystal growth astrophysics: accretion disks geophysics: mantle convection, climate/weather prediction, tsunami simulation,... Module 10: Case Study Computational Fluid Dynamics, Winter 2012/2013 2

3 Part I: Modelling Mathematical Models for CFD Advection and Diffusion Advection Equation Advection-Diffusion Equation Euler Equations 1D Euler Equations Conservation Laws in Higher Dimensions 2D Euler Equations Navier-Stokes Equations Conservation and Convection Form Incompressible Equations Viscous Forces Boundary Conditions Module 10: Case Study Computational Fluid Dynamics, Winter 2012/2013 3

4 Fluids and Flows ideal or real fluids ideal : no resistance to tangential forces compressible or incompressible fluids volume change of gases (vs. liquids?) under pressure viscous or inviscid fluids think of the different characteristics of honey and water Newtonian and non-newtonian fluids the latter may show some elastic behaviour (e.g. in liquids with particles like blood) laminar or turbulent flows turbulence: unsteady, 3D, high vorticity, vortices of different scales, high transport of energy between scales Module 10: Case Study Computational Fluid Dynamics, Winter 2012/2013 4

5 Mathematical Models for CFD typically: all require different models our focus here: incompressible, viscous, Newtonian, laminar incompressible Navier-Stokes Equations Shallow Water Equations starting point: continuum mechanics macroscopic properties (pressure, density, velocity field) compared to stochastic or micro-/mesoscopic approaches (lattice Boltzman method, e.g.) relies on basic conservation laws (remember the heat equation): conservation of mass and momentum (and energy) additionally: slight focus on Finite Volume Methods Module 10: Case Study Computational Fluid Dynamics, Winter 2012/2013 5

6 Advection Equation F(a,t) v(a,t) q(x,t) F(b,t) v(b,t) a b x Conservation of some quantity q in a fluid domain Ω = [a, b] with given velocity v(x, t): b total amount/mass of q in Ω = [a, b] is given by q(x, t) dx change of mass can only happen due to in-/outflow at a and b: b b q(x, t) dx = F (a, t) F (b, t) = F(x, t) b a t = F(x, t) dx x a a note: F (a, t) and F(b, t) denote an inflow into the domain Ω Module 10: Case Study Computational Fluid Dynamics, Winter 2012/ a

7 Advection Equation (2) F(a,t) v(a,t) q(x,t) F(b,t) v(b,t) a b x Consider flux function F (x, t) depends on velocity v(x, t), density q(x, t) and the pipe s cross-sectional area A(x): F(x, t) = A(x)v(x, t)q(x, t) for simplicity, we set A(x) = 1, and obtain: b b q(x, t) dx = t a a b F (x, t) dx = x a ( ) v(x, t)q(x, t) dx x Module 10: Case Study Computational Fluid Dynamics, Winter 2012/2013 7

8 Advection Equation (3) F(a,t) v(a,t) q(x,t) F(b,t) v(b,t) a b x Advection Equation: for smooth functions, we may write: b a t q(x, t) dx = b b ( ) q(x, t) dx = v(x, t)q(x, t) dx t x a a as this equation has to hold for any Ω = [a, b], we demand: ( ) q(x, t) = v(x, t)q(x, t) t x or short: q t + (vq) x = 0 Module 10: Case Study Computational Fluid Dynamics, Winter 2012/2013 8

9 Advection and Diffusion Diffusion even in a fluid at rest, an inhomogeneous density q(x, t) will slowly change towards a uniform density q 0 due to molecular processes diffusion Fick s law of diffusion: resulting flux is prop. to gradient of q F diff = βq x to model both advection and diffusion, we have F = vq + βq x, and thus q t + (vq) x = βq xx special case q t = 0 advection-diffusion equation : βq xx + (vq) x = 0 Module 10: Case Study Computational Fluid Dynamics, Winter 2012/2013 9

10 1D Euler Equations with our quantity q being the mass density ρ, we obtain an equation for the conservation of mass: ρ t + (vρ) x = 0 another conservation property is that of momentum ρv; here, the flux term includes the pressure p: F mom = ρv 2 + p thus, we obtain as equation for the conservation of momentum: (ρv) t + (ρv 2 + p) x = 0 we obtain a system of two PDEs, the 1D Euler Equations to close the system, we need a relation between ρ and p (using the ideal gas law, e.g.) we might add an equation for temperature (derived from the conservation of internal energy) Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

11 Conservation Laws in Higher Dimensions in 2D, a conservation law for quantity q takes the form: or similar in 3D: q t + F (q) x + G(q) y = 0 q t + F (q) x + G(q) y + H(q) z = 0 for advection, the flux functions are F (q) = uq G(q) = vq H(q) = wq where u, v, w are the velocity components in the three space dimensions x, y, z hence, for 2D we obtain a conservation law such as q t + (uq) x + (vq) y = 0 Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

12 2D Euler Equations in 2D, with velocity components u(x, y, t) and v(x, y, t) the equation for conservation of mass reads: ρ t + (ρu) x + (ρv) y = 0 similar, the two equation for conservation of momentum are: (ρu) t + (ρu 2 + p) x + (ρuv) y = 0 (ρv) t + (ρuv) x + (ρv 2 + p) y = 0 again, we assume constant temperature, and we need a relation between ρ and p to close the system the Euler equations model an inviscid (ideal) fluid we also neglect additional source terms, such as for gravity forces, etc. Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

13 Navier-Stokes Equations mass conservation/continuity equation is the same as for the Euler equations: or, written in vector notation: ρ t + (ρu) x + (ρv) y + (ρw) z = 0 ρ + (ρ u) = 0, t u u = x + v y + w z momentum conservation/momentum equations (ρ u) + ( u ρ u) σ f = 0 t with σ being the stress tensor, which includes the pressure p and viscous forces: σ = pi +... f models external (volume) forces (gravity, e.g.) Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

14 Navier-Stokes Equations Conservation and Convection Form the equations for mass and momentum, on the previous slide, are given in the so-called conservation form with the equations (ρ u) = u ρ+ρ u and (ρ u u) = u ( (ρ u) ) +(ρ u ) u, we obtain: ρ + u ρ + ρ u = 0 t t (ρ u) + u( (ρ u) ) + (ρ u ) u σ f = 0 with t (ρ u) = ρ t u + u t ρ and applying u t ρ + u( (ρ u) ) = u ( t ρ + (ρ u)) = 0, we obtain for the momentum equation in convection form ( ) ρ u + ( u ) u σ f = 0 t Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

15 Navier-Stokes Equations Incompressible Equations in the convective forms ρ + u ρ + ρ u = 0 t ( ) ρ u + ( u ) u σ f = 0 t we assume that the density ρ is constant: t ρ = 0, ρ = 0 we obtain obtain the incompressible Navier-Stokes equations: u = 0 ( ) ρ u + ( u ) u σ f = 0 t incompressible : the density does not change due to pressure or temperature, e.g. Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

16 Viscous Forces Open question: stress tensor σ σ includes pressure p and viscosity tensor τ: σ = pi + τ Newtonian fluids: viscous stresses proportional to the strain rate (first derivatives) isotropic, incompressible fluids, Stokes assumption (no volume viscosity), then σ = p + µ u µ the dynamic viscosity Incompressible Navier-Stokes equations: u = 0 ( ) ρ u + ( u ) u = p + µ u + f t Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

17 Dynamic Similarity of Flows Dimensionless Form of the Navier-Stokes Equations we scale our unknowns to typical length scale L and velocity u : x x L t u t L u u u p p p ρu 2 and obtain the dimensionless form of the Navier-Stokes equations: u = 0 t u + ( u ) u = p + 1 Re u + f introducing the Reynolds number Re := µ ρu L important corollary: flows with the same Reynolds number will show the same behaviour Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

18 Boundary Conditions (here only velocity) no-slip: the fluid can not penetrate the wall and sticks to it u = 0. free-slip: the fluid can not penetrate the wall but does not stick to it u n = 0, u n = 0. inflow: both tangential and normal velocity components are prescribed u = u inflow. outflow: should be do nothing ; simple option: all velocity components do not change in normal direction u n = 0. periodic: same velocity and pressure at inlet and outlet u in = u out. Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

19 Part II: A Finite Difference/Volume Method for the Incompressible Navier-Stokes Equations Numerical Treatment Spatial Derivatives Finite Volume Discretisation and Upwind Flux Marker-and-Cell Method, Staggered Grid Discretization of Continuity Equation Discretization of Momentum Equation Time Discretization Chorin Projection Implementation Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

20 Finite Volume Discretisation Advection-Diffusion Equation compute tracer concentration q with diffusion β and convection v: βq xx + (vq) x = 0 on Ω = (0, 1) with boundary conditions q(0) = 1 and q(1) = 0. equidistant grid points x i = ih, grid cells [x i, x i+1 ] back to representation via conservation law (for one grid cell): xi+1 x i x x i+1 F (x) dx = F(x) = 0 x i with F(x) = F (q(x)) = βq x (x) + vq(x). we need to compute the flux F at the boundaries of the grid cells; however, assume q(x) piecewise constant within the grid cells Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

21 Finite Volume Discretisation Advection-Diffusion Equation (2) wanted: compute F(x i ) with F(q(x)) = βq x (x) + vq(x) where q(x) := q i for each Ω i = [x i, x i+1 ] computing the diffusive flux is straightforward: options for advective flux vq: symmetric flux: βq x xi+1 = β q(x i+1) q(x i ) h upwind flux: vq xi+1 = vq(x i) + vq(x i+1 ) 2 vq { vq(xi ) if v > 0 xi+1 = vq(x i+1 ) if v < 0 Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

22 Finite Volume Discretisation Advection-Diffusion Equation (3) system of equations: for all i x i+1 F (x) = F (x i+1 ) F(x i ) = 0 x i for symmetric flux: β q(x i+1) 2q(x i ) + q(x i 1 ) h 2 + v q(x i+1) q(x i 1 ) = 0 2h leads to non-physical behaviour as soon as β < vh 2 (observe signs of matrix elements!) system of equations for upwind flux (assume v > 0): β q(x i+1) 2q(x i ) + q(x i 1 ) h 2 + v q(x i) q(x i 1 ) = 0 h stable, but overly diffusive solutions (positive definite matrix) Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

23 Marker-and-Cell Method Staggered Grid Marker-and-Cell method (Harlow and Welch, 1965): discretization scheme: Finite Differences can be shown to be equivalent to Finite Volumes, however based on a so-called staggered grid: Cartesian grid (rectangular grid cells), with cell centres at x i,j := (ih, jh), e.g. pressure located in cell centres velocities (those in normal direction) located on cell edges Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

24 Spatial Discretisation Continuity Equation: mass conservation: discretise u evaluate derivative at cell centres, allows central derivatives: ( u) i,j = u x + v i,j y u i,j u i 1,j + v i,j v i,j 1 i,j h h remember: u i,j and v i,j located on cell edges notation: ( u) i,j := ( u) xi,j (evaluate expression at cell centre x i,j ) Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

25 Spatial Discretisation Pressure Terms note: velocities located on midpoints of cell edges u v t =... i+ 1 2,j t =... i,j+ 1 2 thus, all derivatives need to be approximated at midpoints of cell edges! pressure term p: central differences for first derivatives (as pressure is located in cell centres) p x p i+1,j p i,j p i+ 1 2,j h y p i,j+1 p i,j i,j+ 1 h 2 Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

26 Spatial Discretisation Diffusion Term for diffusion term u: use standard 5- or 7-point stencil 2D: 3D: u i,j u i 1,j + u i,j 1 4u i,j + u i+1,j + u i,j+1 h 2 u i,j,k u i 1,j,k + u i,j 1,k + u i,j,k 1 6u i,j,k + u i+1,j,k + u i,j+1,k + u i,j,k+1 h 2 Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

27 Spatial Discretisation Convection Terms treat derivatives of nonlinear terms ( u ) u: central differences (for momentum equation in x-direction): u u u i+1,j u i 1,j x u i,j v u i+ 1 2,j 2h y v u i,j+1 u i,j 1 xi+ i+ 1 2,j 1 2h 2,j with v ( ) xi+ = vi,j + v i,j 1 + v i+1,j + v i+1,j 1,j 2 upwind differences (for momentum equation in x-direction): u u u i,j u i 1,j x u i,j v u xi+ 2h y v u i,j u i,j 1 xi+ 1 1 xi+ 2h 2,j 1 2,j 2,j if u i,j > 0 and v xi+ > 0 1,j 2 code for CFD lab will mix central and upwind differences (and is based on conservation form of convection terms) Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

28 Time Discretisation recall the incompressible Navier-Stokes equations: u = 0 t u + ( u ) u = p + 1 Re u + f note the role of the unknowns: 2 or 3 equations for velocities (x, y, and z component) resulting from momentum conservation 4th equation (mass conservation) to close the system; required to determine pressure p however, p does not occur explicitly in mass conservation possible approach: Chorin s projection method p acts as a variable to enforce the mass conservation as side condition Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

29 Time Discretisation Chorin Projection explicit Euler scheme for momentum equation: ( u (n+1) = u (n) + τ p + 1 ( ) ) Re u(n) u (n) u (n) + g Chorin projection compute intermediate velocity that neglects pressure: ( u (n+ 1 1 ( ) ) 2 ) = u (n) + τ Re u(n) u (n) u (n) + g, u (n+1) = u (n+ 1 2 ) τ p u (n+1) needs to satisfy mass conservation: u (n+1) = 0 leads to a Poisson equation for the pressure: ( ) u (n+ 1 2 ) τ p = 0 p = 1 ( u (n+ 1 )) 2 τ thus, system of linear equations to be solved in each time step Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

30 Implementation geometry representation as a flag field (Marker-and-Cell) flag field as an array of booleans: input data (boundary conditions) and output data (computed results) as arrays Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

31 Implementation (2) Lab course Scientific Computing Computational Fluid Dynamics : modular C-code parallelization: simple data parallelism, domain decomposition straightforward MPI-based parallelization (exchange of ghost layers) target architectures: parallel computers with distributed memory clusters possible extensions: free-surface flows ( the falling drop ) multigrid solver for the pressure equation heat transfer or turbulence models Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

32 Part III: The Shallow Water Equations and Finite Volumes Revisited The Shallow Water Equations Modelling Scenario: Tsunami Simulation Finite Volume Discretisation Central and Upwind Fluxes Lax-Friedrichs Flux Towards Tsunami Simulation Wave Speed of Tsunamis Treatment of Bathymetry Data The SWE Code Model and Discretisation Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

33 The Shallow Water Equations h hu + t x hv hu hu gh2 + y huv hv huv hv gh2 = S(t, x, y) Comments on modelling: generalized 2D hyperbolic PDE: q = (h, hu, hv) T t q + x F(q) + G(q) = S(t, x, y) y derived from conservations laws for mass and momentum may be derived by vertical averaging from the 3D incompressible Navier-Stokes equations compare to Euler equations: density ρ vs. water depth h Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

34 Modelling Scenario: Tsunami Simulation The Ocean as Shallow Water?? compare horizontal ( 1000 km) to vertical ( 5 km) length scale wave lengths large compared to water depth vertical flow may be neglected; movement of the entire water column Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

35 Modelling Scenario: Tsunami Simulation (2) Tsunami Modelling with the Shallow Water equations: source term S(x, y) includes bathymetry data (i.e., elevation of ocean floor) Coriolis forces, friction, etc., as possible further terms boundary conditions are difficult: coastal inundation, outflow at domain boundaries Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

36 Finite Volume Discretisation discretise system of PDEs t q + x F(q) + G(q) = S(t, x, y) y with h hu hv q := hu F (q) := hu gh2 G(q) := huv hv huv hv gh2 basic form of numerical schemes: Q (n+1) i,j = Q (n) i,j τ h ( F (n) i+ 1 2,j F (n) i 1 2,j ) τ h ( ) G (n) G (n) i,j+ 1 2 i,j 1 2 where F (n) i+ 1 G(n) 2,j,,... approximate the flux functions F (q) and i,j+ 1 2 G(q) at the grid cell boundaries Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

37 Central and Upwind Fluxes define fluxes F (n) i+ 1 G(n) 2,j,,... via 1D numerical flux function F: i,j+ 1 2 F (n) i+ 1 2 central flux: F (n) i+ 1 2 = F ( Q (n) i = F ( Q (n) i, Q (n) ) i+1, Q (n) ) 1 ( i+1 := 2 G (n) = F ( Q (n) j 1 j 1, ) Q(n) j 2 F ( Q (n) i ) ( (n) ) ) + F Q leads to unstable methods for convective transport upwind flux (here, for h-equation, F (h) = hu): F (n) i+ 1 2 = F ( h (n) i, h (n) i+1 stable, but includes artificial diffusion { ) hu i if u i+ 1 := 2 hu i+1 if u i+ 1 2 i+1 > 0 < 0 Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

38 (Local) Lax-Friedrichs Flux classical Lax-Friedrichs method uses as numerical flux: F (n) = F ( Q (n) i+ 1 i, Q (n) ) 1 ( i+1 := F ( Q (n) ) ( (n) i + F Q 2 2 i+1) ) h ( (n) ) Q i+1 2τ Q(n) i can be interpreted as central flux plus diffusion flux: h ( (n) Q i+1 2τ ) h 2 Q(n) i = 2τ Q(n) i+1 Q(n) i with diffusion coefficient h2 2τ, where c := h τ is some kind of velocity ( one grid cell per time step ) idea of local Lax-Friedrichs method: use the appropriate velocity F (n) := 1 ( F ( Q (n) ) ( (n) ) ) i+ 1 i + F Q i+1 a i h ( (n) Q i+1 ) Q(n) i Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

39 Wave Speed of Tsunamis consider the 1D case t ( h hu ) + ( ) hu x hu gh2 = 0 with q = (q 1, q 2 ) T := (h, hu) T, we obtain ( ) q1 + ( ) q 2 t q 2 x q2 2/q = 0 2 gq2 1 write in convective form: t ( q1 q 2 ) + f x ( q1 ) = 0 q 2 with ( ) ( ) ( ) f f1 /q = 1 f 1 /q = f 2 /q 1 f 2 /q 2 q2 2/q2 1 + gq = 1 2q 2 /q 1 u 2 + gh 2u Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

40 Wave Speed of Tsunamis (2) compute eigenvectors and eigenvalues of f : λ 1/2 = u ± ( ) gh r 1/2 1 = u ± gh and then with f = RΛR 1, where R := (r 1, r 2 ) and Λ := diag(λ 1, λ 2 ), we can diagonalise the PDE: ( ) w1 + Λ ( ) w1 = 0, w = R 1 q t w 2 x w 2 for small changes in h and small velocities, we thus obtain that waves are advected (i.e., travel) at speed λ 1/2 ± gh recall local Lax-Friedrichs method: F (n) := 1 i ( F ( Q (n) i choose a i+ 1 = max{λ k } 2 ) ( (n) ) ) + F Q i+1 a i ( Q (n) i Q (n) ) i 1 Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

41 Shallow Water Equations with Bathymetry h b h hu + hu hu t x 2 gh2 + hv 0 huv = (ghb) x y hv huv hv gh2 (ghb) y Questions for numerics: treat (bh) x and (bh) y as source terms or include these into flux computations? preserve certain properties of solutions e.g., lake at rest Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

42 Shallow Water Equations with Bathymetry (2) Consider Lake at Rest Scenario: at rest : velocities u = 0 and v = 0 examine local Lax-Friedrichs flux in h equation: F (n) = 1 ( ) (hu) (n) i+ 1 i + (hu) (n) i+1 a i+ 1 ( 2 (n) h i ) h(n) i = 0 F (n) F (n) = a i+ 1 ( 2 (n) i+ 1 2 i h 1 i h(n) i ) a i ( h (n) i h (n) i 1) = 0 note: a i± 1 gh and if b i 1 b i b i+1 then h i 1 h i h i+1 2 thus: lake at rest not an equilibrium solution for local Lax-Friedrichs flux Additional problems: complicated numerics close to the shore in particular: wetting and drying (inundation of the coast) Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

43 SWE An Education-Oriented Shallow Water Code Model & Discretisation Simplified setting (no friction, no viscosity, no coriolis forces, etc.): h hu hv hu + hu gh2 + huv = S(t, x, y). hv huv hv gh2 t x y Finite Volume Discretization: generalized 2D hyperbolic PDE: q = (h, hu, hv) T Wave propagation form: t q + x F(q) + G(q) = S(t, x, y) y Q n+1 i,j = Q n i,j t x t y ( ) A + Q i 1/2,j + A Qi+1/2,j n ( ) B + Q i,j 1/2 + B Qi,j+1/2 n. Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

44 SWE An Education-Oriented Shallow Water Code Model & Discretisation Simplified setting (no friction, no viscosity, no coriolis forces, etc.): h hu hv hu + hu gh2 + huv = S(t, x, y). hv huv hv gh2 t x y Flux Computation on Edges: Wave propagation form: Q n+1 i,j = Q n i,j t x t y ( ) A + Q i 1/2,j + A Qi+1/2,j n ( ) B + Q i,j 1/2 + B Qi,j+1/2 n. simple fluxes: Rusanov/(local) Lax-Friedrich more advanced: f-wave or (augmented) Riemann solvers (George, 2008; LeVeque, 2011), no limiters Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

45 SWE An Education-Oriented Shallow Water Code Unknowns and Numerical Fluxes B Q i,j+0.5 A + Q i 0.5,j h ij (hu) (hv) ij ij b ij A Q i+0.5,j B + Q i,j 0.5 Unknowns and Numerical Fluxes: unknowns h, hu, hv, and b located in cell centers two sets of net updates /numerical fluxes per edge: A + Q i 1/2,j, B Q i,j+1/2, etc. Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

46 SWE An Education-Oriented Shallow Water Code Patches of Cartesian Grid Blocks j=ny+1 j=ny j=ny+1 j=ny j=1 j=0 i=0 i=1 i=nx i=nx+1 j=1 j=0 i=0 i=1 i=nx i=nx+1 Spatial Discretization: regular Cartesian meshes; allow multiple patches ghost and copy layers to implement boundary conditions, for more complicated domains, and for parallelization Module 10: Case Study Computational Fluid Dynamics, Winter 2012/

47 References and Literature Course material is mostly based on: R. J. LeVeque: Finite Volume Methods for Hyperbolic Equations, Cambridge Texts in Applied Mathematics, M. Griebel, T. Dornseifer and T. Neunhoeffer: Numerical Simulation in Fluid Dynamics: A Practical Introduction, SIAM Monographs on Mathematical Modeling and Computation, SIAM, Shallow Water Code SWE: Module 10: Case Study Computational Fluid Dynamics, Winter 2012/