UNCONDITIONAL STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH 2 NUMERICAL METHOD

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 5, Number 3, Pages 7 87 c 04 Institute for Scientific Computing and Information UNCONDITIONAL STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH NUMERICAL METHOD ANDREW D. JORGENSON Abstract. Nonlinear partial differential equations modeling turbulent fluid flow and similar processes present special challanges in numerical analysis. Regions of stability of implicit-explicit methods are reviewed, and an energy norm based on Dahlquist s concept of G-stability is developed. Using this norm, a time-stepping Crank-Nicolson Adams-Bashforth implicit-explicit method for solving spatially-discretized convection-diffusion equations of this type is analyzed and shown to be unconditionally stable. Key words. convection-diffusion equations, unconditional stability, IMEX methods, Crank- Nicolson, Adams-Bashforth. Introduction The motivation of this work is to consider the stability of numerical methods when applied to ordinary differential equations ODEs) of the form ) u t) + Aut) Cut) + Bu)ut) = ft), in which A, Bu) and C are d d matrices, ut) and ft) are d-vectors, and ) A = A T 0, Bu) = Bu) T, C = C T 0 and A C 0. Here and denote the positive definite and positive semidefinite ordering, respectively. Models of the behavior of turbulent fluid flow using convection-diffusion partial differential equations discretized in the spatial variable give rise to a system of ODEs, such as 3) u ij t) + b h u ij ɛ 0 h) + ν) h u ij + ɛ 0 h) h u ij = f ij, where h is the discrete Laplacian, h is the discrete gradient, and ɛh) is the artificial viscosity parameter. System 3) is of the form ) and ), where A = ɛ 0 h) + ν) h, C = ɛ 0 h) h, Bu) = b h. In this case the matrix B ) is constant, but in general it may depend on u, and thus the system is allowed to have a nonlinear part. A linear multistep method for the numerical integration a system u t) = F t, u), such as ), is 4) k α j u n j = j= k β j F n j, j= where t is defined on I = [t 0, t 0 + T ] R, u n j R d, F n j = F t n j, u n j ). This work will discuss the regions of stability for implicit-explicit IMEX) methods applied to systems of the form ), and prove that unconditional stability the Received by the editors January 5, Mathematics Subject Classification. 76D05, 65L0, 65M. 7

2 7 A. JORGENSON method s stability properties are independent of the choice of step-size ) holds for a proposed Crank-Nicolson Adams-Bashforth CNAB) IMEX numerical method, u n+ u n + A C) ) 5) AA C) u n+ + A 3 C) A C) un + CA C) un ) + BE n+ )A C) AA C) un+ + A 3 C) A C) un ) + CA C) un ) = f n+, where E n+ = 3 u n + u n, an explicit approximation of ut n+ ). This method is a second-order convergent numerical scheme of the form 4). Section discusses earlier related results for IMEX methods, and Section 3 motivates the unconditional stability analysis of 5) by deriving illustrative stability results for related scalar IMEX methods. With these results in mind, unconditional stability of method 5) is proven in Section 4. Section 5 demonstrates the theory with several numerical tests, the last of which shows the method s effectiveness when applied to a system that is a close variation of 3).. Previous IMEX Stability Results In [5], Frank et al. consider applying IMEX methods to a system of ODEs of the form u t) = F t, ut)) + Gt, ut)), where F is the stiff, and G is the non-stiff parts of the system. Considering the scalar test equation u t) = λut) + γut), they find that under these conditions, λ and γ lying in the regions of stability of their respective methods are sufficient conditions for the IMEX method to be assymptotically stable. As is demonstrated in Section 3, these are not necessary conditions when the system is under assymptions ), which is due to the additional requirement that A C be positive-definite. Akrivis et al. study a system of the same form as ) except that B is assumed to be self-adjoint instead of skew-symmetric. They analyze a general class of methods that are implicit in all linear terms, and explicit in all nonlinear terms, and show these methods to be absolutely stable []. Finally, Anitescu et al. [] show that the first-order IMEX method u n+ u n 6) + Au n+ Cu n + Bu)u n+ = f n+ is unconditionally stable. Unlike in [] there are two linear terms, one of which will be approximated explicitly, while the solution vector in the nonlinear term Bu)ut) is computed using an implicit scheme. 3. Stability for Scalar IMEX Methods 7) 8) Consider the Cauchy problem y t) = ɛ + ν)λyt) ɛλyt), y : R R, y0) =, λ < 0, 0 < ν, 0 < ɛ.

3 STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH METHOD 73 Note that this is the Dahlquist test-problem y t) = νλyt), with exact solution yt) = e νλt, broken into two parts. Definition. A-stability Dahlquist 963)[6]. The multistep method 4) applied to the Cauchy test problem 7) is A-stable if A C where A is entire region of stability for the method). This is equivalent to requiring the numerical solutions u n 0 as t n +. A method s A-stability region can be illustrated by plotting its root locus curve, that is, the values of λν corresponding to the stability boundary roots ζλν) = of its generating polynomials. Recall that for stability the roots of these polynomials must be lie within the unit circle ζ j λν) in modulus) [6]. The aim is to explore IMEX methods which, when applied to the Cauchy test problem 7) as stated, display stable behavior. Let us consider methods which apply an implicit scheme to the first part, and an explicit scheme to the second part. A-priori it is not obvious which mixed methods will exhibit stable behavior if at all), and if so, whether the stability properties of the implicit or explicit part will dominate. 3.. Backward Euler Forward Euler BEFE) IMEX. Let us first investigate an IMEX method which is Backward Euler for the implicit part and Forward Euler for the explicit part: u n+ u n 9) = ɛ + ν)λu n+ ɛλu n. This method can be solved for u n in terms of λ, ɛ, ν, and an initial condition u 0. Iterating backward n times gives the sequence of numerical solutions ɛλ ) n. u n = u 0 ɛ + ν)λ As n +, u n 0 if ɛλ ɛ+ν)λ <. The assumptions in 8) are sufficient for this to hold. None of these conditions is dependent on the choice of step-size, so we can immediately conclude that this method is unconditionally stable. Note that if ɛ is allowed to be zero, we recover the Backward Euler BE) method, which has the solution ) n. u n = u 0 νλ Figure. Energy of BE and BEFE

4 74 A. JORGENSON Figure. Root Locus Curves for BEFE Figure shows the convergence of the energy defined to be ut u) of the solutions of BE and BEFE for initial condition u 0 =, λ = 0000, ν =.00, =.0, ɛ =.0 for the BEFE scheme). Notice that BE converges faster than BEFE mixed method. This illustrates that the advantages of using an IMEX method come at the cost of decreased speed of convergence of the method s solutions. To see the stability region of the BEFE method in terms of step-size and eigenvalues, take ζ n = u n, µ = λν, and solving the method 9) for µ gives the root locus curve 0) µ = ν ρζ) σζ) = ν ζ α + ν)ζ ɛ. Since e iθ = for all θ, taking ζ = e iθ in 0) and letting θ vary in [0, π] produces the desired stability region with ν=.00). The first plot in Figure illustrates that the BEFE IMEX method is stable for any choice of µ outside the solid blue line, which is to say the method 9) is A-stable since any choice of λν in C will be stable and the solution u n will converge to zero as n gets large. The second plot in Figure shows, somewhat counterintuitively, that the stability region of BEFE is growing with ɛ; that is, the region of absolute stability grows as the scaling of the explicit part of the method approaches that of the implicit. We are interested in finding a second-order convergent IMEX method that is also A-stable. Consider ) u n+ u n = ɛ + ν)λ u n+ + u n ) ɛλ 3 u n u n ), which is a Crank-Nicolson second-order implicit) method for the first part of the Cauchy problem 7), and Adams-Bashforth second-order explicit) for the second part. If ɛ is allowed to be zero we recover Crank-Nicolson, which gives: u n = [ + νλ νλ ] n. The characteristic polynomial of method ) is Πr) = ɛ + ν)λ)r + ɛ + ν)λ 3 ɛλ)r ɛλr0 = 0.

5 STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH METHOD 75 This second-degree polynomial has two roots, ɛλ r, = ν) ± 4 + 4λν + νλ ν 8ɛ). λɛ λν Drawing on results from the theory of difference equations, the analytical solutions of the CNAB scalar method ) can be written as u n = γ r n + γ r n. Using initial conditions u 0, u to solve for γ, γ gives u n = + u + r u 0 r + r r n + u + r u 0 r + r r n. Figure 3 shows the convergence of the energy of the solutions of Crank-Nicolson and Crank-Nicolson Adams Bashforth for initial conditions u 0 =, u =.8, λ = 0000, ν =.00, =.5, ɛ =.0. As with BE and BEFE, pure Crank- Nicolson converges faster than the mixed method. Figure 3. Energy comparisons of CN to CNAB 3.. Crank-Nicolson Adams-Bashforth IMEX. For method ) the root locus curve is ) λν = ν ρζ) σζ) = ν ζ ζ ɛ + ν) ζ +ζ ) ɛ 3 ζ ). The first plot in Figure 4 shows the region of stability for CNAB IMEX is similar to that of BEFE IMEX, and this method is also A-stable. The second plot shows the root locus curves corresponding to different values of ɛ. As with BEFE, the region of stability is growing with ɛ. This plot is similar, except for the size of the stability region, for any choice of ɛ 0.

6 76 A. JORGENSON Figure 4. Root Locus Curves for CNAB Figure 5 shows that for for ɛ =.00 and ɛ =.0 the region of stability for BEFE is relatively larger than that of CNAB. This reflects the fact that using a higher order method comes at the cost of a decreased region of stability. Figure 5. Root Locus Curves of BEFE compared to CNAB 3.3. G-stability. Now let us study the stability of the two aforementioned methods from the perspective of a stability definition that is more complex and in some cases more useful. Consider the Lipschitz condition 3) Re F t, u) F t, û), u û) L u û. If the system u t) = F t, u) satisfies 3) with L = 0, then its solutions are contractive. In this case we wish to know which linear multistep methods also have contractive solutions, and are thus G-stable as defined in Definition stated below. Let U n = u n+k, u n+k,..., u n ) T be a sequence of numerical solutions to 4), and define the G-norm of U n to be U n G = U T n GU n.

7 STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH METHOD 77 Definition. G-stability Dahlquist 976)[3]. A multistep numerical method is G-stable if the system of ODEs u = F t, u) satisfies 3) with L = 0, and if there exists a symmetric positive-definite matrix SPD) G, such that 4) U n+ Ûn+ G U n Ûn G, for all steps n and step-sizes > 0, where Ûn is a sequence of solutions for 4) that correspond to different initial conditions than U n. Thus, we can use G-stability to test the behavior of a method where the underlying ODE is linear or nonlinear, providing that it satisfies the Lipschitz condition with L = 0. In the nonlinear case, we have G-stability when the difference of the solutions U n Ûn are not growing in the G-norm G-stability of Scalar Crank-Nicolson Adams-Bashforth. Showing that a method is G-stable requires checking that the conditions of the G-stability definition hold. Since our underlying ODE 7) is linear, we can consider the Lipschitz and G-norm conditions 5) Re F t, u), u) 0, U n+ G U n G, respectively. It is easy to see that if λ < 0 and ν > 0 then νλu, u = νλu 0, and the Lipschitz condition is satisfied. Thus, the task is to see if we can construct a G that satisfies the G-stability definition. The G-matrix corresponding to this method can be generated directly or indirectly using the proof Dahlquist s equivalence theorem as is done in Subsection 3.4. [4] Direct Computation of G. First, consider the inner-product ɛ + ν)λ u n+ + u n ) ɛλ 3 u n u n ), u n+ u n 6) ) 0, which holds because they are the RHS and LHS of method ) under consideration. Multiplying by and expanding gives 7) where 8) E = c u n+ c c )u n+ u n + c u n c 3 u n+ u n + c 3 u n u n 0 c = ɛ + ν)λ, c = ɛ + ν)λ 3 ɛλ, c 3 = ɛλ. Now consider the equation E = U n+ G U n G + a u n+ + a u n+ + a 0 u n, a 0, a, a R. Imposing E 0 implies U n+ G U n G, since a u n+ +a u n+ +a 0 u n 0. Let ) g g 9) G=. g g Thus, if the matrix G produced by matching the coefficients of 7) to those of 8) is SPD, method ) is G-stable by Definition. Following this approach and letting g = g produces the following nonlinear system of six equations in six unknowns: 0) u n+ : c = g + a, u n+ u n : c c = g + a a u n : c = g g + a u n+ u n : c 3 = a a 0 u n : 0 = g + a 0 u n u n : c 3 = g + a a 0.

8 78 A. JORGENSON Solving this system produces the G-matrix ) G= λ ɛ ν ɛ ). 4 ɛ ɛ This matrix is symmetric by construction, and it is easy to see that if λ < 0 all its principle minors have a positive determinant, and therefore this G is positivedefinite by Sylvester s Criterion. Thus, by Definition this IMEX method is G- stable as well as A-stable, as demonstrated in the previous section). This, as Dahlquist was finally able to prove in 978, is not a coincidence. Theorem. Dahlquist 978)[4]: If a method s generating polynomials ρ, σ, have no common divisor, then the method is G-stable if and only it is A-stable Constructing G Using Generating Polynomials. By following the proof of Theorem see [4], [6]) one can derive a procedure for computing the G matrix that relies on algebraic manipulation of the the method s generating polynomials rather than solving a nonlinear system as is required for computing G directly. The generating polynomials for scalar CNAB IMEX are Define the function which for CNAB is ρζ) = ζ ζ, σζ) = ɛ + ν) ζ +ζ ) ɛ 3 ζ ). Eζ) = ρζ)σ ζ ) + ρ ζ )σζ)), Eζ) = ζ ζ)α + ν) ζ + ζ ) ɛ 3 ζ )) ζ ) + ζ ζ )ɛ + ν) ζ +ζ ) ɛ 3 ζ ))) ɛζ ) = 4ζ [ ɛ ][ = [ζ ɛ ] ) ] ][ ζ ) ] = aζ)a ζ ). Define the function P ζ, ω) = ρζ)σω) + ρω)σζ)) aζ)aω), which with some simplification and factoring becomes P ζ, ω) = 4 [ ɛζ ) ω ) + ɛω )ω ζαν 4ɛ)ω + 3αω ) This yields the matrix ) + ζ ɛ 3ɛω + ɛ + ν)ω ))] ɛ + ν) = ζω ) ζω ɛ 4 4 ζ ɛ 4 ω + ɛ ) 4 = ζω )g ζω g ζ g ω + g ). ) G= ɛ + ν ɛ, 4 ɛ ɛ which is SPD. Multiplying ) by the positive constant λ gives the same result as computing G directly as in the previous section.

9 STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH METHOD 79 Somewhat surprisingly, under assumptions 8) the method s stability properties are driven by its implicit part, and this is the motivation that leads to the separation of linear parts A and C as done in ). 4. Unconditional Stability of the Crank-Nicolson Adams Bashforth IMEX Method. Let us now return to considering system ) under assumptions ). The CNAB method proposed in Section is a member of a broader family of three level, second order time-stepping schemes [8]: 3) θ + )u n+ θu n + θ )u n + A C) AA C) θun+ + θ)a θ + )C ) A C) un ) + CA C) θun ) + BE n+θ )A C) AA C) θun+ + θ)a θ + )C ) ) A C) un + CA C) θun ) = f n+θ, where E n+θ is an implicit or explicit second order approximation of u n+ and θ [, ]. Taking θ =, and E n+θ = 3 u n u n, gives method 5). Here we will consider the stability properties of the method 5). However, unlike the examples in the previous chapter, the system under consideration will be d- dimensional and be in terms of non-commmuting coefficient matrices A, B, C, where B is allowed to be nonlinear. This added complexity is worthwhile since many processes have highly nonlinear behavior, but it comes at the cost of greatly complicating stability analysis. 4.. Well-Posedness of the Problem. Nonlinearity of problem ) will prohibit the Lipschitz condition 3) from holding globally. This is not trivial, since it means none of the well-known global stability results for the underlying problem will necessarily hold for a discussion of the well-posedness of globally Lipschitz continuous Cauchy problems see [7], Chapter 0). The system is, however, locally stable. Theorem. Local Stability of Nonlinear System ). Under assumptions ), the solution of ) is bounded as ut) u0) e t + F T e t ), is therefore stable for all t I = [0, T ], for all finite T. where F T = max t [0,T ] ft), For the proof of Theorem see []. This result ensures that problem ) is wellbehaved locally, which is to say that its exact solutions ut) do not blow up on I. This allows us to conclude that the problem is sufficiently well-posed in at least a local sense, and we can discuss stability of a numerical method for approximation of its solution on I. 4.. Transformation of the Method. In [] the numerical solution u n provided by the BEFE IMEX method 6) is shown to be nonincreasing, u n+ E u n E, E = I + C, in the energy norm E, and this condition is sufficient to conclude the method is unconditionally stable. The aim of this section is analogous. Borrowing heavily from

10 80 A. JORGENSON the G-stability concepts developed in Section 3.3, and given an appropriately chosen transformation of the method, it can be proved that the G-norm of the numerical solutions are decreasing at each time step, and the method is unconditionally stable on interval I. Since Bu) is assumed to be skew-symmetric, multiplying method 5) from the left by the vector [ A C) AA C) un+ + A 3 C) A C) un )] T + CA C) un ) will cause the nonlinear term BE n+ )A C) AA C) un+ + A 3 C) A C) un ) + CA C) un ) to disappear, leaving A C) AA C) un+ + u n+ u n = A C) + A C) ) A 3 C) A C) un + CA C) un ), AA C) un+ + A 3 C) A C) un AA C) un+ + A 3 C) A C) un ) + CA C) un ) + CA C) un ) ), f n+. By the properties of the inner-product and Euclidian norm, this can be rearranged as u n+ u n, A C) AA C) u n+ + A 3 C) A C) un ) + CA C) un ) + AA C) u n+ + ) A 3 C) A C) un + CA C) un ) = f n+, A C) AA C) u n+ + A 3 C) A C) un 4) ) + CA C) un ). Focusing on the first line of 4), the goal will be to simplify the transformed method into positive pieces using the G-norm to group and compare terms, as was done in the G-stability examples in Section 3.4. The G-stability matrix is calculated using the procedure derived from the proof of Theorem, as demonstrated in Subsection Method 5) and its corresponding characteristic polynomials yield matrix A C) G= A ) 4 C)A C) A C) 4 C)A C) 5). A C) 4 C)A C) A C) 4 C)A C) Referring to the G-stability examples in Section 3.4, taking A = α + ν)λ, C = αλ, and ignoring A C) terms, G matches matrix ).

11 STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH METHOD G-norm. We now wish to check that G is a symmetric positive-definite matrix so it can be used to finish putting the transformed method 4) into norms and positive terms Symmetry of G. The G matrix defined in 5) is a block-partitioned matrix with submatrices of size d d. Since the off-diagonal blocks are the same, symmetry of the four blocks is sufficient to conclude G is symmetric also. Since A and C are both symmetric by assumptions ), adding or subtracting positive-definite multiples also results in symmetric matrices. By properties of diagonalizable matrices, A C) is symmetric. This is sufficient to conclude that each block of G is symmetric, and therefore G is also Positive-Definiteness of G. If G is positive-definite the following will be strictly positive for any choice of d-dimensional vectors u, v not equal to zero: 6) [ v] u = Lemma. G is a positive-definite matrix. Proof. Expanding equation 6) gives G [ [ u u, G. v] v] [ u [ u [, G = v] v] 4 u T [A C) A C)A C) ]u 7) u T [A C) CA C) ]v v T [A C) CA C) ]u + v T [A C) CA C) ]v ]. Subtracting and adding 4[ u T [A C) CA C) ]u ] to 7) gives [ u [ u [, G = v] v] 4 u T [A C) A C)A C) ]u 8) u T [A C) CA C) ]u + u T [A C) CA C) ]u u T [A C) CA C) ]v + v T [A C) CA C) ]v ]. The first two terms can be combined and simplified to be 4 ut [A C) A C)A C) ]u = ut u, which is the energy of the method s solutions. Take F = A C) CA C). Since F = F T, the remaining terms can be factored as 4 [ut F ) T F u u T F ) T F v + v T F ) T F v] = 4 F u F v, F u F v = 4 F u F v 0, and the result immediately follows.

12 8 A. JORGENSON 4.4. Unconditional Stability Result. As proved above, the matrix G is symmetric and positive-definite, and therefore the expression defined in 6) is a G- norm. Lemma. Let u n satisfy 5) for all n {,..., T }. Then u n+ u n, A C) AA C) un+ + A 3 C) A C) un ) + CA C) un ) = ] [ ] un u n G u n [ un+ G + 4 u n+ u n +u n ) F Energy Bound. The proof of Lemma allows us to conclude that [ v] u 9) ut u > 0, G that is, the energy of the solutions is bounded from above by their G-norm, and this is independent of. From 9) we have convergence of the solutions in the G-norm implies convergence of their energy Energy Equality. To see that the method is unconditionally stable consider the following energy equality, which holds for u 0, u inital conditions at all time steps n = through N : [ ] un u N + N n= [ u G = ] + u 0 G 30) + N u n+ u n + u n F 4 n= AA C) un+ + A 3 C) A C) un + CA C) un ) N n= f n+, A C) AA C) un+ + ) A 3 C) A C) un + CA C) un ). Notice Lemma and the Energy Equality 30) immediately imply G-stability in the case of ft) = 0. That is, if the the energy source forcing function) is removed, stability of the method requires that the G-norm of the solutions decays weakly at each time step n. To see that this note that [ ] un+ [ ] un + ) u u n G u n+ u n +u n n G 4 F + AA C) un+ + A 3 C) A C) un + CA C) un ) = 0 holds for all n {, N }. Further, ) u n+ u n +u n 4 0, F since F is a positive-definite matrix. Thus we have [ ] un+ [ ] un. u n u n G Since this result is independent of the the size of time-step, we have unconditional stablility when ft) = 0. G

13 STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH METHOD Energy Estimate. When ft) 0 for some t I, the effect of f n+ on the Energy Equality 30) is ambiguous. By applying Cauchy-Schwarz and Young s Inequalities, the Energy Bound 9), and combining like-terms, we can use 30) to derive the following energy estimate to bound the effect of f on the energy in the system: u N + + N n= N n= u n+ u n + u n F AA C) un+ + A 3 C) A C) un + CA C) un 3) [ ] u N u + A C) fn+. 0 G n= Inequality 3) gives that solutions u N are bounded from above by the G-norm of initial conditions and a positive term depending on f, so although monotonicity is no longer required, boundedness is retained. 5. Numerical Tests Consider the general form of the linear multistep numerical method 4). Method 5) is of this form, where k = and α = I, α 0 = I, α = 0 β = A C) AA C) BEn+ )A C) AA C) β 0 = A C) A 3 C) A C) BEn+ )A C) A 3 C) A C) β = A C) CA C) BEn+ )A C) CA C). Applying this method to system ) will require solving for the vector u n+ in terms of u n, u n given two initial condition vectors u 0, u ), that is u n+ = [ ] [ [I ] ] 3) I hβ + hβ0 un + hβ u n + hf n+. The method requires the inversion [ I + ha C) AA C) + hben+ )A C) AA ] C). In practice 3) will not be solved by computing the inverse since this would be overly costly and introduce large round-off error. Also of note is that in general, A, B, and C do not commute, and this plays a critical role in the stability analysis developed in the previous section. To demonstrate that the proposed CNAB method 5) is unconditionally stable consider the following numerical experiments. 5.. Test. Take ) ) A = ɛ + ν), C = ɛ, 0 0 BE n+ ) = ) 0 00 u + u, ft) = 0, 00 0

14 84 A. JORGENSON Figure 6. Convergence CNAB with ft) = 0 where u and u denote the first and second elements of the vector 3 u n u n not the time step). Let ν =.00, and initial conditions be u = u = [, ] T. Figure 6 shows the convergence of the energy and G-norms for CNAB 5) with d =, and ɛ =.0. Notice that as in the scalar example in Section 3 see Figure 4), the G-norm decreases monotonically, even though the energy of the solution does not. The second plot shows the convergence of the G-norm for CNAB 5) with d = and ft) = 0 for various. Figure 7. Convergence CNAB with ft) = e t for = 0.5, Convergence CNAB with ft) = e t for various 5.. Test. Now we relax the restrictions on C and ft) to study the case where C is not diagonal, and ft) 0. Taking C = ɛ ) 00, ft) = e t, implies that ) 00ν ɛ A C =. ɛ ν

15 STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH METHOD 85 Recalling that A C is required to be positive-definite, by Sylvester s Criterion we see that A C is positive-definite when 00ν ɛ > 0, which is satisfied by all ν and ɛ such that 0ν > ɛ. ɛ =.009 satisfies the inequality for ν =.00. Figure 7 shows the energy and G-norms for various under the new conditions on C and ft). Notice that when = 0.5 the G-norm is not monotically decreasing until t. This is due to the forcing function ft) = e t, and is an illustration of the Energy Estimate 3), which says that the solutions in the G-norm are bounded by solutions at previous steps and a norm depending on the forcing function ft). The Energy Bound 9) holds in both experiments, as the theory requires Convection-Diffusion Equation. Finally, we test method 5) on an equation that is commonly used to model turbulent fluid flow. Consider the convectiondiffusion partial differential equation with two spatial variables defined on Ω = [0, ] [0, ], where u t + b u ɛ u = f over Ω, u = φx) on δω, ux, 0) = u 0 x) in Ω. In [] Anitescu et al. use this equation to test their first-order method, and here we will do the same for the second-order scheme. Discretizing the spatial variables and using central differences and averaging for the antidiffusion operator gives 33) u ij t) + b h u ij ɛ 0 h) + ν) h u ij + ɛ 0 h) h u ij q = fij. This is of the form ), and is the same system as 3), except C is now the matrix that results from the ɛ 0 h) h u ij q term, where q is the number of times the averaging operation is applied. This results in a C matrix that is postive-definite but not sparse. Let b = cosθ), sinθ)), θ = 7, and the inital conditions be f = 0 and u = 0 for the PDE). We take q = and the averaging operation to be an equal weighted average of nearest neighbors. Since the method is second-order and two previous time-steps are required for every step, the first step is calculated using the firstorder scheme proposed in []. For the boundary conditions we take a line of angle θ through the center of Ω and let φ = on the boundary above this line and φ = 0 on boundary below it. For all tests we use a 3 3 uniform mesh for the spatial variables Spatial Solution. Figure 8 shows the numerical solution for ɛ 0 = 0.000, and ɛ 0 = 0., both with 000 steps, and a step-size of 0. In [] it is observed that the steady-state solution for the first order scheme applied to 33) is sensitive to the choice of artificial viscosity paramter ɛ 0. In our tests we do not observe this sensitivity.

16 86 A. JORGENSON Figure 8. Spatial solution for two choices of artifical viscosity parameter 5.4. G-norm of the Solution. Figure 9 shows the G-norm of the solution to 3) for time-steps of 0. and. The non-zero boundary conditions give a non-zero ft) for ), and in this case the G-norm of the solution may not decrease montonically. Figure 9 suggests that u 00 is a good approximation of the steady-state solution. Since test problem 3) is linear, we can consider its homogenous form by subtracting from ), giving A C + B)u 00 = ft), u t) + A C + B)ut) u 00 ) = 0. The Energy Equality 30) requires that u n u 00 G be a montonically decreasing sequence, and this is demonstrated in Figure 0. This, together with the Energy Bound 9) demonstrates the convergence of the numerical solution given by the second-order method 5) for this test problem. Figure 9. Convergence of the solution in the G-norm

17 STABILITY OF A CRANK-NICOLSON ADAMS-BASHFORTH METHOD 87 Figure 0. Numerical Validation of the Energy Equality 30) Acknowledgments The author thanks the anonymous referees who offered useful comments, Kiersten Bethke for her patience and support, and Catalin Trenchea for his many suggestions for improving this work. References [] Akrivis, G., Crouzeix, M., and Makridakis, C., Implicit-Explicit Multistep Finite Element Methods for Nonlinear Parabolic Problems, Math. Comp., ), [] Anitescu, M., Pahlevani, F., and Layton, W., Implicit for Local Effects and Explicit for Nonlocal Effects is Unconditionally Stable, ETNA, 8 988), electronic). [3] Dahlquist, G., Error Analysis for a Class of Methods for Stiff Non-linear Initial Value Problems, Springer-Verlag, New York, 976. Vol. 506 of Lecture Notes in Mathematics. [4] Dahlquist, G., G-Stability is Equivalent to A-Stability, BIT Numerical Mathematics, 8 978), [5] Frank, J., Hundsdorfer, W., and Verwer, J. G., On the Stability of Implicit-Explicit Linear Multistep Methods, Appl. Numer. Math., 5 997) [6] Hairer, E., and Wanner, G., Solving Ordinary Differential Equations II, Springer-Verlag, Berlin, 00. Vol. 4 of Springer Series in Computational Mathematics, Second Revised Edition. [7] Quarteroni, R., and Saleri, F., Numerical Mathematics, Sprinter-Verlag, Berlin, 007. Vol. 37 of Texts in Applied Mathematics, Second Edition. [8] Trenchea, C., Second Order Implicit for Local Effects and Explicit for Nonlocal Effects is Unconditionally Stable, submitted 0). Department of Mathematics, Western Washington University, Bellingham, WA 985, USA andrew.jorgenson@wwu.edu