Lecture Notes on Numerical Schemes for Flow and Transport Problems

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1 Lecture Notes on Numerical Schemes for Flow and Transport Problems by Sri Redeki Pudaprasetya sr Department of Mathematics Faculty of Mathematics and Natural Sciences Bandung Institute of Technology ITB Summer School 07: Mathematical Modelling on Industrial and Environmental Problems Institut Teknologi Bandung, Bandung, Indonesia -5 July 07

2 Contents Introduction Finite Difference Method for Diffusion Equation 3. The Euler explicit method Stability Backward time center space Theta-method and the Crank-Nicolson method Finite Difference Method for the Convection Equation 7 3. Upwind method Finite Volume Method for Conservative Problems 0 4. Examples on the approximation for f(u) The Riemann Problem Finite volume method for Burgers equation Introduction In this course you will learn numerical methods, mainly finite difference method for the linear partial differential equations, and a little bit on finite volume method for the nonlinear equations. We will also discuss ways to handle boundary conditions: Dirichlet as well as Neumann types. We hope that this will give you a strong background in dealing with various problems that needs numerical solutions. In principal, finite difference is a method to approximate a differential with a ratio of differences. Application of finite difference approximation to differential equation will give us a difference equation. Basic finite difference formulas are as follows: Forward difference: x f x f(x + x) f(x ) x Backward difference: Center difference: x f x f(x ) f(x x) x x f x f(x + x) f(x x) x + O( x), () + O( x), () + O( x ). (3) Task: Apply Taylor expansion of f(x) around x, and derive all finite difference formulas above. There are several important properties of difference equations for partial differential equations that must be considered before numerical computations are performed. For a well-posed differential equation with an analytical solution u(x, t), and the corresponding difference equation with numerical solution u(x, t n ). Relation between differential equation and the corresponding difference equation is the concept of consistency. Relation between the analytical solution u(x, t) and the approximate solution u(x, t n ) is the concept of convergence. These connections are illustrated in the diagram in Figure. The modified differential equation (MDE) can be obtained if we express each terms in the difference equation, in a Taylor series at a base point. Effectively, this changes the difference equation back into a partial differential equation. If the MDE tends to the original partial differential equation as grid spacing tends to zero, the difference equation is called consistent. Terms appearing in the MDE that do not appear in the original partial differential equation are called as

3 Figure : A connection diagram between the differential equation and its difference equation counterpart. the local discretization error or truncation error terms. Order of the local discretization error is the order of the difference scheme. Here are explanation of some basic terminology of a numerical scheme: The order of a difference equation is the order of the local discretization error. A difference equation is consistent with the partial differential equations if the difference between them vanishes as the size of x, t tend to zero. A difference equation is stable if it produces a bounded solution. A finite difference method is convergent if the solution of the difference equation approaches the exact solution of the partial differential equation. Lax equivalence theorem: It states that for a consistent finite difference method for a wellposed linear initial value problem, the method is convergent if and only if it is stable. Proving convergence of a numerical solution is often difficult, it is much easier showing stability and consistency of a difference scheme. Therefore in practice, Lax equivalence theorem is often used. In summary, the concepts of consistency, order, stability and convergence must be considered when solving a partial differential equation by finite difference methods. Consistency and order can be determined from the modified differential equation. Stability can be investigated by applying the von Neumann method. Convergence can be ensured through Lax equivalence theorem by checking consistency and stability. Finite Difference Method for Diffusion Equation In this section, we will discuss common methods of finite difference for the diffusion equation. Consider a diffusion equation with thermal diffusivity κ u t = κu xx, for 0 < x < L, t > 0, (4) (Boundary condition) u(0, t) = 0, u(l, t) = 0, for t > 0, (5) (Initial condition) u(x, 0) = f(x), for 0 < x < L. (6) In this section, we will discuss various finite difference scheme, starting from the simplest FTCS which is explicit and of order one, the BTCS implicit method, and finally the order two, the Crank-Nicolson scheme and theta method. 3

4 . The Euler explicit method On the computational domain Ω = { (x, t) R 0 < x < L, t > 0 }, we introduce a grid, with x denotes the spatial grid size and t the time step. The grid points (x, t n ) are defined as x = ( ) x, =,,, N x +, x = L/N x, (7) t n = (n ) t, n =,, (8) We discretize (4) by replacing the time derivative t u(x, t n ) by the forward difference approx- Figure : Stencil of the Euler explicit (or FTCS) method for the diffusion equation. imation, and the space derivative u(x x, t n ) by the center difference approximation. Writing the numerical approximation of u(x, t) at grid points (x, t n ) as u n, we obtain the difference equation of (4) as t (un+ u n ) = κ x (un + u n + u n ). (9) The scheme can be simplified as u n+ = u n + S(u n + u n + u n ), with S = κ t x. (0).. Stability To analyze the stability of the FTCS scheme, consider a numerical solution of the form u n = e ikx iωt n, () with k is the wave number and ω the wave frequency. Different representation is u n = λ n e ikx, with λ = e iω t. () Substituting () into the scheme (0) will give us λ = + S(e ik x + e ik x ) = 4S sin (k x/) Thus at each time step, the Fourier mode e ikx is amplified by the factor λ. Finite solution u n for all n will be obtained if λ, which is satisfied if S = κ t x. (3) Task: prove that (3) is the stability condition of the Euler explicit scheme (0). We stress here that condition (3) is the necessary condition for the stability of (9). 4

5 Exercise:. As mention before, Local discretization error is the residual d n found as the difference between the differential equation (4) evaluated at grid point (x, t n ) and the scheme (0). Show that d n = κ ( κ t 6 x) u 4x n, which is of the order O( t, x ). Then, discuss the consistency of (0) as the discrete equation of (4).. Write a numerical code for the explicit Euler equation (0). Solve the diffusion equation (4) on 0 < x < with κ =, using boundary conditions (5) and initial condition f(x) =. For computation, use x = 0., t = 0.005, and plot the numerical solution u(x, t = ). Compare the numerical result with the analytical solution (as obtained form the separation variables method). 3. The FTCS method has an interesting feature when S = 6. In this special case the O( t) and O( x ) of the local discretization error d n cancels exactly, and the method becomes O( t, t x, x 4 ). Check the improved accuracy from your numerical results, by comparing the error of numerical solution u(x, ), at several positions, i.e. x = L/4, L/, 3L/4. Aside: Condition (5) is of Dirichlet type. This condition is easily satisfied by taking u n = 0, and u n N x+ = 0, for all n. And we simply implement (0) for =, 3,, N x. Another type of boundary condition that often met is the Neumann boundary condition, which is as follows u x (0, t) = 0, u x (L, t) = 0, for t > 0. (4) Implementation of this boundary conditions needs extra attention. Apply the order-two center difference to approximate (4), from which we can get the relation u( x, t n ) = u( x, t n ). Using this, we can implement (0) for =, for all n. Next, the right boundary condition u x (L, t) = 0 can be solved in analogous way. Note that, in this Neumann boundary conditions case, values u n and un N x+ should be computed, at all time step tn. % (Algorithm for diffusion equation u t = κu xx ) % Initialition % Specify the computational domain, spatial 0 x L and time 0 t T. % Choose the grid size x and t. % Compute N x + and N t, the total number of grid points in x and t. % Define an initial condition u(x, 0). u( : N x +, ) = 0; u(0.5 N x 4 : 0.5 N x +4, ) = ; % This is a square wave initial condition, centered in the middle. % Specify the value of C Courant number. C = κ t/ x ; % Stability condition: Courant number C /. % Computing u(x, t n+ ) using the difference equation (??). for n = : N t u(, n + ) = 0 for = : N x u(, n + ) = u(, n) + S (u(, n) u(, n) + u(, n)); end u(n x +, n + ) = 0 end 5

6 % Plotting the result xplot=0: x:l for n=:nt figure() plot(xplot, u(:,n+)) pause(0.) drawnow end. Backward time center space Implementation of the backward approximation for time derivative t u(x, t n+ ) and center difference approximation for x u(x, t n+ ) will give us the following implicit scheme t (un+ u n ) = κ x (un+ + un+ + u n+ ). (5) The scheme can be simplified as Su n+ + + ( + S)un+ Cu n+ ) = un, with S = κ t x. (6) Figure 3: Stencil of the BTCS implicit method for the diffusion equation. Exercise: The numerical properties of the implicit scheme (6), and its implementation for solving (4-6).. Show that the local discretization error is d n = κ ( κ t x) u 4x n, which is of order O( t, x ).. Show that the scheme (6) is unconditionally stable. 3. In order to compute u n, for every time step, we need to solve a system of equation Aun+ = u n, with ( + S) S 0 0 S ( + S) S 0 A m m = S ( + S) The vector u n has u n as its element. What is the size of the square matrix A, is it N x, N x or N x +? (Remember that if we deal with Dirichlet boundary condition (5), we already have u n = 0 and un N x+ = 0 for all n.) 4. Write a numerical code for solving (4-6) using the implicit scheme (6). Try several combination of x and t to make sure that the BTCS is an unconditionally stable scheme. 6

7 .3 Theta-method and the Crank-Nicolson method In this section we will move on to the second order scheme. This second order scheme is a convex combination of the explicit scheme and implicit scheme. For a parameter θ, with 0 θ, a combination of θ implicit and ( θ) explicit will result in the following scheme t (un+ u n ) = κ θ x (un + u n + u n ) + κ θ x (un+ + un+ + u n+ ). (7) The scheme (7) with θ = is called the Crank-Nicolson scheme. This is a second order scheme with accuracy O( t, x ). Moreover, it is an unconditionally stable scheme. Figure 4: Stencil of the theta-method. Exercise: Numerical properties of the θ scheme (7). Find the local discretization error d n, and show that (7) is an order two scheme with accuracy O( t, x ).. Show that the scheme (7) is unconditionally stable. 3 Finite Difference Method for the Convection Equation In this section we will discuss numerical methods for the equation u t + du x = 0, subect to the initial condition u(x, 0) = f(x), with x R. We recall that the exact solution of this initial value problem is given by u(x, t) = f(x dt). This solution represents a signal propagates undisturbed in shape with velocity d. The signal propagates to the right if d > 0, and to the left if d < 0. We will use this analytical solution as a benchmark for the numerical schemes. Note that these schemes are easily generalized for more complex problems (but analysis are not as easy as this linear case). 3. Upwind method For discussions in this sub section, we address the following problem u t + du x = 0, for 0 < x < L, t > 0, (8) (Initial condition) u(x, 0) = f(x), for 0 < x < L. (9) Here, we restrict to the case of d > 0 and discuss the simplest finite difference scheme, which is the upwind method. On the computational domain Ω = { (x, t) R 0 < x < L, t > 0 }, we introduce a grid, with x denotes the spatial grid size and t the time step. The grid points (x, t n ) are defined as x = ( ) x =,,, N x +, x = L/N x, (0) t n = (n ) t n =,, () 7

8 For the case d > 0 in which signal propagates to the right, upwind method is Forward Time Backward Space (FTBS). Applying FTBS to the convection equation (8) will yield u n+ or explicitly u n t + d un un x = 0. () u n+ = ( C)u n + Cu n, (3) with C d t x a Courant number. From the implementation of forward time and backward space approximation, we can guess already that this upwind method has an accuracy O( t, x). To support this argument, you should show that the local discretization error of the upwind approximation is indeed of the order O( t, x). Please try! Figure 5: Stencil for the upwind method. From the initial condition (9), values of u, for =,,, N x + is known. The FTBS scheme (3) can be used to compute u n, for =,, N x +, for all t n, n =,,, but not u n, because (3) will need un 0 which is not defined. Here, we need to take a numerical boundary condition: u n, for all n. For the first try, one can ust take un = 0, for all n. Later on you can experiment with various left boundary u n = g(tn ), with g(t) plays role as a signal enters from the left boundary. By doing this, we can compute u n, for =,, N x +, for all n using the FTBS scheme (3). % (Algorithm for convection equation u t + du x = 0) % Initialition % Specify the computational domain, spatial 0 x L and time 0 t T. % Choose the grid size x and t. % Compute N x + and N t, the total number of grid points in x and t. % Define an initial condition u(x, 0). u( : N x +, ) = 0; u( : 4, ) = ; % This is a square wave initial condition. % Specify the value of C Courant number. C = d t/ x; % Stability condition: Courant number C. % Computing u(x, t n+ ) using the difference equation (3). for n = : N t u(, n + ) = 0;% Zero left boundary for = : N x + u(, n + ) = ( C)u(, n) + Cu(, n); end end 8

9 % Plotting the result figure() surf(u ) % A surface plot in Matlab command, compare with surf(u) drawnow Exercise. Numerical aspect of the upwind FTBS scheme. Derive the stability condition of the FTBS method. Show that the local discretization error is d(d t x)u xx n, which is of order one.. Write a numerical code for the FTBS scheme (3). For the first simulation take a combination of t and x such that the Courant number C = d t x is exctly one, since d > 0. Using this set up, you should get a square wave propagates to the right. Make sure your wave propagates with the correct velocity d. Also check that the wave influx that enters from the left side of the domain matches with your chosen numerical left boundary. 3. Perform another simulation using different t in such a way that C =., explain the result. Since the Courant number is larger than one, your numerical solution should be unstable, in which the numerical value u n increases as time progresses. 4. Conduct another computation using different t in such a way that the Courant number C is strictly less than one. Your simulation should demonstrate that very soon discontinuity of the square wave disappear, due to diffusion effect. This is a direct consequence of the error diffusive term d(d t x)u xx n. This damping effect will only be observed when d t x 0. For a stable scheme, this damping effect will be observed only when C = d t x strictly less than one. Please check this statement using your numerical code. Exercise. Another idea is to implement the forward time and center space (FTCS) approximation. This scheme seems preferable for two reasons: it may accommodate both cases d > 0 and d < 0, and it has second order accuracy in space: FTCS scheme is of order O( t, x ). Implement stability analysis and argue that this FTCS scheme is always unstable, which means this scheme is useless.. Upwind scheme for arbitrary d. t (un+ u n ) = d+ x (un u n ) d x (un + u n ), (4) with d + max(d, 0) dan d min(d, 0). Implement this scheme using the initial condition: a hump with peak located in the middle of the domain. Take a positif value of d and simulate the propagation of the initial hump to the right. Then, without changing the scheme, take the negative value of d, and show that the initial hump is now propagates to the left. Note: higher stable scheme are Lax O( t, x ), Lax-Wendroff O( t, x ), and leapfrog O( t, x ). Interested reader may consult reference [3, ] for further study. 9

10 4 Finite Volume Method for Conservative Problems In this section we will discuss another variant of numerical method, which is called the finite volume method. Finite volume methods are closely related to finite difference methods, and a finite volume method can often be interpreted directly as a finite difference approximation to the differential equation. However, finite volume methods are derived on the basis of the integral form of the conservation law, a starting point that turns out to have many advantages. Whereas in finite difference methods all derivatives appear in the equation, each is approximated by a ratio of differences. On the computational domain [0, L] which is divided into N x cells of homogeneous length x, a staggered partition points are x / = 0, x 3/ = x,, x +/ = x,, x Nx+/ = N x x = L (5) We assume that u(x, t n ) denotes the averaged value of u in cell C = [x, x + ]. Within time interval t, this value changes according to the net flux from the left and right boundaries of cell C according to ( u(x, t n+ ) x = u(x, t n ) x t f(x ), t n ) f(x +, t n ). (6) Equation (6) is exactly the conservative principle in discrete form of the quantity u on cell C. What we need to do next is ust a good approximation scheme for flux f(x, t n ). For further discussion, we use the following notations u n = u(x, t n ), and f n = f(x, t n ), for all x and t n in the computational domain. 4. Examples on the approximation for f(u). If the flux f(u) = κu x, equation (8) reduces to diffusion equation u t = κu xx. For the approximation, we take an average f n + (f n + + f n ), the discrete equation (6) reduces to u n+ u n = κ t x (f n + f n + f n ), (7) which is exactly the finite difference approximation (0).. If the flux f(u) = du, equation (8) reduces to convection equation u t + du x = 0. Here we take the upwind approximation: if d > 0, f + f = du, and when d > 0, f + f + = du +. Implementing this on (6) will give us the finite difference approximation (4). Prove this! 4. The Riemann Problem We start the discussion by addressing the basic conservation equation of variable u with flux function f(u) as follows u t + f(u) x = 0, for x R, t > 0, (8) { ul, if x < 0, u r, if x > 0. (9) u(x, 0) = 0

11 Writing b(u) = f (u), equation (8) can be written as u t + b(u) u = 0, for x R, t > 0. (30) x Equation (30) is written in the form of convection equation with b(u) represents the signal velocity. Some observation, if u(x, t) is a solution, then so is u(ax, at), for arbitrary a > 0, implying that the solution is a similarity solution of the form u(x, t) = û(x/t). Solutions of (8,9) are propagated along the characteristic x/t = b(u), with b(u) = f (u) represents the velocity. For the initial condition (9), we can distinguish two cases, b(u l ) > b(u r ) and b(u l ) < b(u r ), to be discussed below. Case b(u l ) > b(u r ). The characteristic emanating from x < 0 have a smaller slope than the slope of the characteristic coming from x > 0. As a consequence, there is an area in which the characteristic intersect, which would lead to multi valued solutions. Here, we have a discontinuous solution. We can easily verify that the following is a solution of the Riemann problem (8,9), and it is given by u(x, t) = { ul, if x/t < s, u r, if x/t > s, (3) where s is defined by the Rangkine Hugoniot condition s = f(u r) f(u l ) u r u l (3) Solution of (3) is known as the shock wave, and s as the shock speed. This solution contains discontinuity, and it is considered as weak solution. A typical shock wave solution and the corresponding characteristic are shown in Figure 6. The shock speed formula (3) is determined such that (3) is a solution of the corresponding integral form of (8), see [] for further study. Figure 6: Shock wave and the corresponding characteristic. Case b(u l ) < b(u r ). In this case the characteristic emanating from x < 0 have a larger slope than those emanating from x > 0, so we have separating characteristic, see Figure 7 (right). In this case we do not expect a discontinuous solution. Solution of (30) is given by u l, if x/t < b(u l ) u(x, t) = w(x/t), if b(u l ) < x/t < b(u r ) (33) u r, if x/t > b(u r ).

12 Figure 7: Rarefaction wave and the corresponding characteristic. where w(η) should satisfy b(w(η)) = η. This solution is called rarefaction wave, and its typical profile is depicted in Figure (7) (left). Despite the fact that the initial condition (9) is discontinuous, the rarefaction wave (33) is continuous. Please check! Exercise:. Burgers equation is the (8) with the flux f(u) = u, written explicitly as follows u t + ( ) x u = 0, subect to the piecewise constant initial condition { α, if x < 0, u(x, 0) = β, if x > 0. In this case b(u) = f (u) = u, therefore two type of solutions are distinguish by the relation between b(α) = α and b(β) = β. If α > β, the solution is a shock wave, formulate this shock wave, and determine the shock speed. If α < β, the solution is a rarefaction wave, formulate the solution.. The kinematic LWR model for traffic flow is equation (8) with the chosen flux. The most common flux function used is the Greenshield flux, in which after normalization f(u) = u( u). Similarly, there are two type solutions of the kinematic wave equation u t + (u( u)) = 0, x subect to the initial condition { α, if x < 0, u(x, 0) = β, if x > 0. In this case b(u) = f (u) = u, therefore two type of solutions are distinguish by the relation between b(α) = α and b(β) = β. If α < β, the solution is a shock wave, formulate this shock wave, and determine the shock speed. If α > β, the solution is a rarefaction wave, formulate the solution.

13 4.3 Finite volume method for Burgers equation Consider the Burgers equation which is the conservative equation with f(u) = u, written explicitly as u t + ( ) x u = 0, x R, t > t n. (34) Basically, the most important step in implementing the finite volume method (7) is on the approximation of f n. The approximation, which is based on analytical solution, holds for + each cell V = [x /, x +/ ] in the computational domain, see Figure 8. Therefore, we first discuss the analytical solution of (34) with the following initial condition { u u(x, t n n ) =, x < x +/ u n +, x > x +/. (35) This analytical formula will be used to find the suitable app Figure 8: Sketch of the spatial domain in the finite volume method, and illustration of the piecewise constant numerical solution at time level t n. If u n > un +, the solution is a shock wave propagating with speed s n = (un + u n +). Let us denote η = x x + t t n. Then, the shock wave solution is given by u R (η; u n, u n +) = { u n, if η < s n u n +, if η > sn. (36) 3

14 If u n < un +, the solution is a rarefaction wave as given by u n, if η < un u R (η; u n, u n +) = η, if u n < η < un + (37) u n +, if η > un +. By taking the numerical flux f x + = u R (η; un, un + ), we obtained the following (Show this!) If u n > un +, then f n + = { (un ), if s n > 0 (un + ), if s n < 0 (38) If u n < un + f n + = (un ), if u n > 0 0, if u n < 0 < un + (un + ), if u n + < 0 (39) We resume here that the finite volume method is (7) with flux given by (38,39), this scheme is also known as Godunov method. Exercise: Figure 9: Shock wave solutions at time t = calculated using Godunov method (left) and Upwind method (right), in comparison with the analytical solution.. Solve the Burgers equation subect to the initial condition {, if x < 0, u(x, 0) =, if x > 0. Analytical solution will be a shock wave, with discontinuity that propagates with speed s =.5. In this case the scheme reduces to u n+ = u n t ( x (un ) ) (un ) (40) Implement this Godunov method, and write a numerical code to simulate the propagation of a shock wave solution. Use x =.5 0, x =.5 0 and plot numerical solution together with the analytical solution at time t =. 4

15 . Alternatively, Burgers equation can be written as u t + uu x = 0. (4) Implementation of the upwind scheme, and since u > 0 we obtain the following scheme u n+ = u n t ( x un u n u n ). (4) Write a numerical code to simulate the propagation of a shock wave solution. Use x =.5 0, x =.5 0 and plot numerical solution together with the analytical solution at time t =. It is shown in (9) that the upwind method cannot capture the shock speed correctly. Clearly the Godunov method (40) is much better compare to the upwind scheme (4). The Godunov scheme can capture the speed of the shock wave. This is one of characteristic features of a conservative scheme. Moreover, the scheme (4) is considered as the non-conservative scheme. References [] Matthei, R.M.M., Rienstra, S.W., ten Thie Boonkkamp, J.H.M., Partial Differential Equations, Modelling, Analysis, Computations, SIAM Monographs, Philadelphia, 005 [] Strauss, W., Partial Differential Equations, an Introduction, John Wiley & Sons, 99. [3] Hoffmann, J., Numerical Methods for Engineers and Scientists, Mc.Graw-Hill, 99. 5