topic4: Implicit method, Stability, ADI method

topic4: Implicit method, Stability, ADI method
AE/ME 339 Computational Fluid Dynamics (CFD) K. M. Isaac Professor of Aerospace Engineering 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Implicit form of difference equation In the previous explicit method, u i,n depends only on u i-1,n-1 , u i,n-1, and u i+1,n-1 , all which are at time level n-1. Nature of solution in explicit method. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR In this formulation, solution at (i,n), ui,n is affected only by the values along and below boundary PQR (region A) in the previous figure. Values in region above PQR (region B) do not influence ui,n. Exact solution u(x,y) at Q depends on the values at all times earlier than tn, a property of parabolic PDE. The CFL criterion is a limitation of the explicit method. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Fully Implicit Method Illustrated by the figure below 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR As before, crosses (x) denote grid points used for and circles (o) for The equation now becomes Rearranging 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR IC and BC are the same as before. Equations similar to the above should be written for each grid point Note: left boundary has i=0 and right boundary has i=M, thus total of (M+1) grid points (also knows as nodes) are present. Thus we have (M-1) linear simultaneous equations with (M-1) unknowns. Explicit solution is not possible. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Convergence of Implicit form Can show using Taylor series Where utt and uxxxx are evaluated at (i , n+1). 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR From the above It can be shown that implicit method converges to the exact solution of the PDE as and for any value of The difference equation is now written for as follows. For 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Finally, = 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Note: In the matrix shown above, change in subscript N=M-1, adopted for simplicity. The RHS terms d1,d2,…dN are known quantities. The matrix above is called a tridiagonal matrix i.e., only sub-diagonal, diagonal, and super diagonal terms are non–zero. Solution can be obtained by Gauss elimination. Recursion solution Constants bi and gi are to be determined. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Substituting into the ith equation of the set for ui-1 gives Rewrite as 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Comparing the two equations we have recursion relations for b and g From the first equation Where and 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR From the last equation 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Substitution in the last equation yields Algorithm summary i=n-1, n-2,…,1 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Recursion formulas for and i=1, 2, 3,…,N i=1, 2, 3,…,N Gauss elimination can cause large round-off errors. Implicit schemes usually require more computational steps, but the ratio Dt/(Dx)2 has no restrictions, is a definite advantage. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Stability (7.10) A finite difference form is convergent if the solution tends to the exact solution as (in the absence of round off error). Stability refers to amplification of information present in IC, BC or introduced by errors in the numerical procedure such as round off error. Von-Neumann’s stability analysis: Stability implies only boundedness, not the magnitude of deviation from the true solution. Key features of stability analysis: Assume that: i) At any stage, t=0 here, a Fourier expansion can be made of some initial function f(x), and a typical term in the expansion can be written as ejbx where b is a positive constant and 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR ii) Separation of time and space variable can be made. At time t, the term becomes By substituting in the difference equation, the form of y(t) can be determined and stability criterion established. Example Explicit finite difference form Substitute for each u. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Cancel exp (jbx) throughout Where l = Dt/(Dx)2 Trig. identity : Can be used to get 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Since If we choose y(0) = 1, this has the solution And can be proven by substitution. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR For stability, y(t) must be bounded as This requires An amplification factor x is usually defined as follows: 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Which shows for stability. In the Fourier expansion we considered only one term corresponding to 1 value of b. When all possible values of b are considered sin(bDx/2) could become 1. Therefore, the stability condition becomes Intuitively it implies that ui,n affects ui,n+1 in a “non-negative” manner. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR A similar analysis for the implicit method would give Since for all l, the procedure is unconditionally stable. Consistency means that the procedure may in fact approximate the solution of the PDE under study and not the solution of some other PDE. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Crank-Nicolson method (7.12) Previous explicit and implicit methods have discretization error Recall: 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Recall the central difference operator Let us now try the following form for the second derivative 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR The above form involves 6 points to represent And q obeys the following: Depending on the value of q, the method will be explicit (q = 0), implicit (q = 1), or a combination of the two. For the Crank–Nicolson (C-N) method, q = ½. The difference equation now becomes C-N method has the following properties: i) Stable for all values of the ratio, l = Dt/(Dx)2 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR (ii) Has truncation error When written in full,the equation becomes Dufort-Frankel Method (7.13) Method is an unconditionally stable, explicit method 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR 3 time levels are involved More difficult to formulate IC More computer storage is required Error 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Alternating-Direction Implicit (ADI) Method (7.14) The unsteady state heat conduction in a slab is governed by the following equation Top and bottom surfaces are Insulated BC are imposed on the 4 sides Figure 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Explicit Method Stability Criterion: Implicit Method Writing in full with yields 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Scheme is stable for all values of λ There are 5 unknowns per equation Gauss elimination for solution is more complicated System is not tri-diagonal ADI Method Let us now consider a parabolic PDE in two dimensions denoted by x and y i.e., 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR ADI uses two finite difference equations used in turn over successive time steps each of size The first equation is implicit only in the x-direction Second equation is implicit only in the y-direction is an intermediate value at the end of time step Step 1 Note that there is no time subscript for 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Step 2 values are solved for in the first step and values are solved for in the second step Advantage is that the matrices in both steps are still tri-diagonal Exercise: Write the equations in full using and Can be shown that procedure is unconditionally stable Discretization error 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR ADI can also be used for solving elliptic PDE’s ADI is not recommended for 3D problems Example 7.3 Infinitely long bar has: Thermal diffusivity Square cross section of side 2a IC: Temperature is uniform at T0 BC: side surface temperature T1 Figure Compute temperature distribution T(x,y,t) inside the slab 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Can write Procedure Non-dimensionalize the equations as follows Observe: Problem has symmetry in geometry, IC and BC about both x and y axis 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Need to solve only one quadrant Due to symmetry there is no heat flux across X, Y axes (insulated boundaries) IC: BC: throughout the domain along sides X=1 and Y=1 along X=0 along Y=0 figure 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Types of BC (7.17) Instead of u , boundary , or a combination may be specified at the u=g Dirichlet condition: Neumann condition: Combination: Where α, β, γ are constants and g is a known function Consider the following BC from heat transfer at the straight boundary x = 0 (see figure CLW 7.9) 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Consider the earlier parabolic PDE and may be obtained as before For ,use Taylor series as follows Using the BC we get 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Write the corresponding equation for uxx for the heat conduction problem with an insulated boundary. 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Final implicit form of FD approximation (2D parabolic) at point (0,j) Example: 1D heat conduction problem with insulated end BC at insulated end is Therefore from the above equation (set a=g=0) 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR At point (i = 0) equation becomes ………………………(A) From (A) 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Non –linear PDE’s The heat conduction equation of the previous sections is linear Fluid flow equations often have non-linear terms Example: x-Momentum equation of 2D steady, incompressible flow Since u and v are the velocity components in x,y directions respectively the LHS terms are non-linear Previous techniques can be adapted to solve non-linear equations The basic approach is to linearize the equations 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR In , if the coefficient u of is treated as a known quantity, then the equation becomes linear When unsteady equations are solved u at the beginning of the time step can be used as the multiplier For example, the first term can be discretized as Would be the fully implicit form of the first term when we use the forward difference form for 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR Note that superscript n denotes quantities at time level tn,, which would be known from the previous solution step Exercise: Write the same for the 2nd term When steady state problems are solved using iterative techniques, values from the previous iteration step would be used as the multiplier u Other non-linear forms Consider , the mass diffusion term Note D( c ), the diffusion coefficient, is a function of the dependent variable c, the concentration 2/22/2019 topic4: Implicit method, Stability, ADI method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR If we use the model the above term becomes The first term on the RHS would be linearized as before using as the multiplier To use the implicit procedure for the 2nd RHS term, it can be split as and treat the first half as a constant. Note α and β are constants in the above discussion 2/22/2019 topic4: Implicit method, Stability, ADI method

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topic4: Implicit method, Stability, ADI method

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