Finite Volume Schemes: an introduction

Transcription

1 Finite Volume Schemes: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato in Scienze dell Ingegneria Civile e Ambientale Corso di Metodi Numerici marzo 2008

2 Outline Hyperbolic Conservation Laws Classical and Weak solutions The Riemann problem Finite Volume Methods in 1D Finite Volume Methods in 2D The lectures about Finite Volume and Mixed Hybrid Finite Element will be on line at A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

3 Hyperbolic Conservation Laws Finite Volumes (FV) schemes are most useful for modelling hyperbolic conservation laws, which, in their simplest (scalar) form may be expressed by the PDE u t + f = 0 on a domain Ω, with u(x, t) specified on inflow boundaries. Example: the equation for conservation of mass in groundwater contamination problem. Assumptions: no diffusion, no mechanical dispersion, density of the contaminant and the water constant, velocity of the fluid mixture constant, Let ρ(x, t) be the density of the contaminant at point x and time t and v(x, t) be the velocity of the fluid mixture. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

4 The total pollutant mass in two sections 1 and 2 is given by x 2 x 1 ρ(x, t) dx. If no source or sinks are present, the mass can change only because of the mixture flowing across the endsections in x 1 or x 2. The rate of mass change is given by: d dt x2 x 1 ρ(x, t) dx = ρ(x 1, t)v(x 1, t) ρ(x 2, t)v(x 2, t) By integrating in time from t k to t k+1 we obtain the integral form of the conservation law. Assuming ρ and v differentiable, with simple calculations we obtain the differential form of the conservation law: ρ t + (ρv) x = 0 A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

5 Classical and Weak solutions Solution of hyberbolic conservation laws may be obtained if initial and boundary conditions are given. The simplest problem is the initial value problem, or Cauchy problem, defined for < x < and t 0. We must specify initial conditions only: u t + f = 0 u(x, 0) = u 0 (x) < x <. Classical solutions of this problem are constant along the characteristics curves defined by x(t) such that dx dt = f (u(x(t), t)) t 0 x(0) = x 0 A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

6 Shock waves u is constant along these characteristics and the characteristics travel at constant velocity which is equal to f (u 0 (x 0 )). Generally, classical solutions exist only for t [0, T ] where T 1 = inf x u 0 (x)f (u 0 (x)). At the time t = T the characteristics first cross, the function u(x, t) has an infinite slope the wave is said to break by analogy with waves on a beach and a shock forms. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

7 Weak solutions To allow discontinuities, which arise in a natural way in this situation, we define a weak solution of conservation law. Definition (Weak solution) A function u(x, t), bounded and measurable, is called a weak solution of the conservation law, if for each φ C (R R + ), the following equality holds: + 0 [φ t u + φ x f (u)] dx dt = + φ(x, 0)u(x, 0) dx. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

8 The Riemann problem Definition (The Riemann problem) A Riemann problem is the conservation law together with initial data consisting of two constant states separated by a single discontinuity, { ul x < 0, u 0 (x) = u r x > 0. Example: The Riemann problem in Burgers equation u t + ( 1 2 u2 ) x = 0 A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

9 If u l > u r, there is a unique weak solution, { ul x < st u(x, t) = x > st. s = (u l + u r ) is the shock speed 2 If u l < u r, there are infinitely many weak solutions, since between the points u l t < x < u r t, there is no information available from the characteristics. To determine the correct physical behavior we adopt the vanishing viscosity approach by adding the term ɛu xx depending on small parameter ɛ u r u t + ( 1 2 u2 ) x = ɛu xx. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

10 If the initial data is smooth and ɛ very small then, before the wave begins to break, the ɛu xx term is negligible compared to other terms and the solutions to the two PDEs look nearly identical. As the wave begins to break, the term u xx grows much faster than u x and at some point the ɛu xx term is comparable to the other terms and begins to play a role. This term keeps the solution smooth for all time, preventing the breakdown of solutions that occurs for the hyperbolic problem. The weak solution is called rarefaction wave or expansion fan: u l x < u l t u(x, t) = x/t u l t x u r t x > u r t u r A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

11 u l u r Shock wave Rarefaction wave u r u l 0 0 A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

12 Some common numerical schemes To solve numerically hyberbolic conservation laws some common numerical schemes are: Finite Differences Finite Volumes Finite Elements Discontinuous Galerkin We will see more in detail (but not too much...) the Finite Volume approach. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

13 Properties of the ideal scheme The ideal scheme for the resolution of hyperbolic conservation laws would reflect the properties of the underlying physical and mathematical system: Conservative: for correct capturing of discontinuities; Accurate: obvious...; Free of spurious oscillations: spurious oscillations have no physical meaning. This goal is remarkably difficult to achieve; Continuous: for convergence to the steady state solution. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

14 The Finite Volume method In the Finite Volume method the three main steps to follow are: Partition the computational domain into control volumes (or control cells) - wich are not necessarily the cells of the mesh. Discretize the integral formulation of the conservation laws over each control volume (by applying the divergence theorem). Solve the resulting set of algebraic equations or update the values of the dependent variables. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

15 FV in 1D We will see two FV schemes in 1D: Godunov s method and van Leer s method. Godunov s method uses the exact solution of the local Riemann problem and does not produce oscillations around discontinuities, but it is only first order accurate and the solutions display large numerical viscosity. van Leer s method can be viewed as a generalization of Godunov s method, it is a high resolution method, characterized by second order accuracy on smooth solutions and the absence of spurious oscillations. Both methods are conservative. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

16 Conservative method Definition Given a uniform grid with time step t and spatial mesh size x, a numerical method is said to be conservative if the corresponding scheme can be written as: v n+1 j = vj n λ(g n g n ), j Z n 0 j+ 1 j where vj n approximates u(x j, t n ) at the point (x j = j x, t n = n t), λ = t x and g : R2k R is a continuous function, called the numerical flux (function), that defines a (2k + 1)-point scheme. g n j+ 1 2 = g(v n j k+1,..., v n j+k). A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

17 Motivation of the conservative scheme v n j can be view as an approximation of u n j = 1 x xj+1/2 x j 1/2 u(x, t n ) dx. u n j is the average of u(, t n ) on the cell [x j 1/2, x j+1/2 ] (where x j±1/2 = x j ± x 2 ). From the integral form of the conservation law, we have: xj+1/2 x j 1/2 xj+1/2 u(x, t n+1 ) dx = u(x, t n ) dx [ t n+1 t n x j 1/2 f (u(x j+1/2, t)) dt t n+1 t n f (u(x j 1/2, t)) dt]. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

18 Dividing by x and using u n j we get u n+1 j = u n j 1 t n+1 x [ f (u(x j+1/2, t)) dt t n t n+1 t n f (u(x j 1/2, t)) dt]. The numerical flux function can be considered as an average flux through x j+1/2 over the time interval [t n, t n+1 ], g n j+1/2 = 1 t t n+1 t n f (u(x j+1/2, t)) dt. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

19 Godunov s scheme 1. Projection step Given data vj n at time t n, construct a piecewise constant function ˆv j n (x, t n ) defined by ˆv n j (x, t n ) = v n j x j 1/2 x x j+1/2. v_(j 1) v_(j+1) v_j x_(j 1) x_j x_(j+1) A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

20 2. Evolution step Solve the local Riemann problem at the cell interfaces, that is, on each subinterval [x j, x j+1 ] and for t t n, solve ˆv n j t + n f (ˆv j ) x{ ˆv n j (x, t n ) = = 0 vj n, x j < x < x j+1/2 vj+1 n x j+1/2 < x < x j+1 3. Projection step Define the approximation v n+1 j at time t n+1 by averaging the Riemann problem solution ˆv j n at the time t n+1, so that v n+1 j = 1 x xj+1/2 x j 1/2 ˆv n j (x, t n+1 ) dx. These values are then used to define new piecewise constant data ˆv n+1 j (x, t n+1 ) and the process repeats. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

21 Example In the simple case of a linear advection equation u t + au x = 0, with a > 0, the third step gives v n+1 j = v n j a t x (v j n vj 1 n ). u_(j 1) u_j < > a t u_(j 1) < > a t x_(j 1) x_j x_(j+1) A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

22 Stability condition The CFL (Courant-Friedrichs-Lewy) condition λ max f (u) 1 u 2, where λ = t is a necessary stability condition stating that x the domain of dependence of the method includes the domain of dependence of the PDE. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

23 van Leer s method First and third steps of Godunov s methods are of a numerical nature and can be modified without influencing the physics, in order to define spatially high order accurate scheme. van Leer s scheme is a spatially second order accurate scheme (known also as MUSCL (Monotone Upstream-centered Scheme for Conservation Laws) method). The straightforward replacement of the first-order scheme by a second-order accurate interpolation leads to the generation of oscillations around discontinuities. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

24 To overcome this limitations, non linear components are introduced. This concept was introduced by van Leer under the form of limiters, i.e. functions that control the gradient of the computed solution to prevent the appearance of unphysical over/under shoots. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

25 1 Reconstruction step: the dependent variable is interpolated using a piecewise linear function ˆv starting from the cell averages vj n, vj 1 n, v j+1 n. To this end define Sj n to be the slope on the jth cell calculated using vj 1 n and v j+1 n. Then v ˆv n j n + (x x j ) Sn j x j 1/2 < x < x j+1/2, x (x) = ˆv n (x j 1/2 ) x x j 1/2 ˆv n (x j+1/2 ) x x j+1/2. Note that taking Sj n = 0 for all j and n recovers Godunov s method; A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

26 2 Evolution step: the waves are propagated across cell interfaces according to an exact or approximate solution of a local Riemann problem that uses the interpolated values ˆv n (x) as initial conditions. One solves t w + x f (w) = 0 x R, t n t t n+1 w(x, t n ) = ˆv n (x). This step yields w(, t n+1 ). 3 Cell-averaging step: v n+1 j is obtained by projecting the solution w(x, t n+1 ) onto the piecewise constant functions v n+1 j = 1 xj+1/2 w(x, t n+1 ) dx. x x j 1/2 A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

27 a) b) x j-1 x j x j+1 x x x j-1 j j+1 v t c) d) x x x j-1 j j+1 x x x j-1 j j+1 a) reconstruction; b) limiting; c) evolution; d) cell-averaging step. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

28 Slope limiter The numerical scheme is completely defined after specification of the slope limiter S n j. Poor choice of slopes may give oscillations in the solution. Generation of oscillations can be prevented by acting on their production mechanism and introducing nonlinear correction factors, so called the limiters, that force the method to be total variation diminishing (TVD). Definition A numerical method to solve hyperbolic conservation laws is called total variation diminishing if + j= v n+1 j+1 v n+1 j + j= v n j+1 v n j A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

29 To obtain a slope limiter, let S n j = Ŝn j Φ n j where Ŝn j represents the actual slope approximation and Φ j = Φ(θj n ) is a limiter function, defined in such a way that the method is TVD and θj n is the ratio of two consecutive gradients, i.e.: θj n = v j n vj 1 n vj+1 n v. j n A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

30 Limiter functions Various limiter functions can be found in literature. For example: van Leer s limiter: Φ(θ) = θ + θ 1 + θ, minmod limiter that represents the lowest boundary of the second-order TVD region: { min(θ, 1) if θ > 0 Φ(θ) = 0 if θ 0. the superbee limiter, that represents the upper limit of the second-order TVD region and has been introduced by Roe: Φ(θ) = max[0, min(2θ, 1), min(θ, 2)]. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

31 Numerical example with periodic boundary conditions u t + u x = 0, with initial condition u 0 (x) periodic of period 2 defined on the interval [ 1, 1] as 1 1 x < x < 0.25 u 0 (x) = x < x < x 1 We solve this problem at time t = 2 s with λ = 0.5, by setting t = s and x = m. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

32 1.0 (a) Godunov scheme 1.0 (b) van Leer scheme van Leer limiter (c) van Leer s scheme minmod limiter 1.0 (d) van Leer s scheme superbee limiter A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

33 Numerical example with non periodic boundary conditions u t + u x = 0, with the following initial condition: { sin(πx) 0 x 2 u(x, 0) = u 0 (x) = 0 otherwise. We solve this problem at time t = 2 s with λ = 0.5 and x = m. A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34

34 1.0 (a) Godunov s scheme 1.0 (b) van Leer s scheme van Leer limiter (c) van Leer s scheme minmod limiter 1.0 (d) van Leer s scheme superbee limiter A. Mazzia (DMMMSA) Finite Volume a.a. 2007/ / 34