Spectral method for the unsteady incompressible Navier Stokes equations in gauge formulation

Report no. 04/09 Spectral method for the unsteady incompressible Navier Stokes equations in gauge formulation T. W. Tee I. J. Sobey A spectral method which uses a gauge method, as opposed to a projection method, to decouple the computation of velocity and pressure in the unsteady incompressible Navier Stokes equations, is presented. Gauge methods decompose velocity into the sum of an auxilary field and the gradient of a gauge variable, which may, in principle, be assigned arbitrary boundary conditions, thus overcoming the issue of artificial pressure boundary conditions in projection methods. A lid-driven cavity flow is used as a test problem. A subtraction method is used to reduce the pollution effect of singularities at the top corners of the cavity. A Chebyshev spectral collocation method is used to discretize spatially. An exponential time differencing method is used to discretize temporally. Matrix diagonalization procedures are used to compute solutions directly and efficiently. Numerical results for the flow at Reynolds number Re = 000 are presented, and compared to benchmark results. It is shown that the method, called the spectral gauge method, is straightforward to implement, and yields accurate solutions if Neumann boundary conditions are imposed on the gauge variable, but suffers from reduced convergence rates if Dirichlet boundary conditions are imposed on the gauge variable. Subject classifications: AMSMOS: 65M70, 76M22, 76D05 Oxford University Computing Laboratory Numerical Analysis Group Wolfson Building Parks Road Oxford, England OX 3QD E-mail: Wynn.Tee@comlab.oxford.ac.uk May, 2004

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3 Introduction Spectral methods use high-order global functions, such as trigonometric and algebraic polynomials, to approximate solutions of ordinary and partial differential equations, unlike finite difference and finite element methods which use low-order local functions. If the solution being approximated is sufficiently smooth, then spectral methods yield exponential convergence rates, in contrast to finite difference and finite element methods which yield only algebraic convergence rates. If the solution has only a finite number of nonsingular derivatives, then spectral methods converge only algebraically, though some order of accuracy may be recovered by a suitable treatment of the singularities. The three main classes of spectral methods are spectral Galerkin, spectral tau and spectral collocation methods. The reader is referred to [5] for an overview of spectral methods. Spectral methods, therefore, provide a viable alternative to finite difference and finite element methods for solving ordinary and partial differential equations. In particular, much work has gone into developing spectral methods for solving the unsteady incompressible Navier Stokes equations [6, 9, 20], which are the governing equations of motion for incompressible viscous fluid flows. The unsteady incompressible Navier Stokes equations in primitive variables, or equivalently velocity-pressure formulation, for flow in Ω R d, d being the spatial dimension, with closed boundary Ω, are u t + u u + p = Re 2 u, in Ω,. u = 0, in Ω,.2 where u is the velocity, p is the pressure, and Re is the Reynolds number. The task of solving equations. and.2, subject to initial and boundary conditions for u, is complicated by the coupling between u and p, which arises because p acts as a Lagrange multiplier for the incompressibility constraint. Projection methods, introduced by Chorin [7], decouple the computation of u and p at the expense of imposing artificial boundary conditions on p, resulting in numerical boundary layers and reduced convergence rates for p [0]. Despite these drawbacks, spectral methods for solving the unsteady incompressible Navier Stokes equations continue to use projection methods. Gauge methods, introduced by E and Liu [2], overcome the issue of artificial boundary conditions for p, by writing the unsteady incompressible Navier Stokes equations in a form which does not involve p explicitly. Writing u = a + φ, yields a t + u u + p + φ t Re 2 φ = Re 2 a, in Ω,.3 a + 2 φ = 0, in Ω,.4 where a is an auxilary field, and φ is called the gauge variable. Since a and φ are nonphysical variables, imposing arbitrary conditions on φ, changes a according to the compatibility condition u = a + φ, and so the physical nature of the problem remains

4 unchanged. Imposing p = φ/ t + Re 2 φ, yields a t + u u = Re 2 a, in Ω,.5 2 φ = a, in Ω,.6 which is a form of the unsteady incompressible Navier Stokes equations, called the gauge formulation. The main advantage of the gauge formulation over the usual velocity-pressure formulation, is the freedom to impose boundary conditions on φ. This may be seen as a substitute for imposing artificial boundary conditions on p. Corresponding to the boundary condition u = u Γ on Ω, two possible sets of boundary conditions for a and φ are φ = 0, u Γ τ = a τ, u Γ η = a η + φ, on Ω,.7 η and φ η = 0, u Γ τ = a τ + φ τ, u Γ η = a η, on Ω,.8 where τ and η are unit vectors in the anticlockwise tangential and outward pointing normal directions respectively. Corresponding to the initial condition u = u 0 at time t = 0, initial conditions for a and φ may be prescribed in a similar manner. Equations.5 and.6, and boundary condition.7 form a system known as a Dirichlet gauge formulation. Equations.5 and.6, and boundary condition.8 form a system known as a Neumann gauge formulation. E and Liu [, 2] presented finite difference and finite element discretizations of Dirichlet and Neumann gauge formulations for various flow problems. Wang and Liu [22] showed that the backward Euler time discretization of Dirichlet and Neumann gauge formulations, with explicit boundary conditions for a, respectively yield half-order and first-order convergence for u. Nochetto and Pyo [9] derived similar results under more realistic regularity assumptions, and showed that the backward Euler time discretization of Dirichlet gauge formulations, with explicit boundary conditions for a, yields not only half-order convergence for u, but also no convergence for p, because the numerical approximation of φ does not always satisfy a compatibility condition. In this paper, we present a spectral method for solving Dirichlet and Neumann gauge formulations. The test problem is formulated in Section 2. The spatial and temporal discretization schemes are described in Section 3. The numerical results are compared to benchmark results in Section 4. Finally, conclusions about the method, which we call the spectral gauge method, are made in Section 5. 2 Problem Formulation The lid-driven cavity flow is a standard test problem for validating incompressible flow solvers. The flow is governed by the unsteady incompressible Navier Stokes equations,

5 which are solved in the square cavity Ω = [0, ] [0, ], subject to the initial condition u u, v = 0, 0, and the boundary conditions u, v =, 0 on the wall y =, and u, v = 0, 0 on the other three walls. At the top corners of the cavity, where u is discontinuous, the vorticity ω = u/ + v/ x and p are infinite [5]. At the bottom corners of the cavity, the second derivatives of ω and p are infinite [8]. A suitable treatment of these singularities, such as that provided by a subtraction method [4], is necessary to preserve the high accuracy of solutions computed by a spectral method. Subtraction Method The application of a subtraction method to the lid-driven cavity flow, involves a removal of the leading terms in the asymptotic expansions of u, v and p near the corners of the cavity, so that the modified problem has a less singular solution. The solution of the lid-driven cavity flow is decomposed such that u = u A + u D + ū, v = v A + v D + v and p = p A + p D + p, where u A, v A, p A and u D, v D, p D denote leading terms in the asymptotic expansion of u, v, p at steady-state near the corners A0, and D, respectively, and ū, v, p denotes a less singular solution to be computed by a spectral method. The effect of the singularities at corners B0, 0 and C, 0 is much weaker, and will be ignored. The formulae for u A, v A and p A are derived by considering the steady-state streamfunction ψ, defined by u = ψ/ and v = ψ/ x, which satisfies ψ 4 2 ψ ψ = Re ψ 2 ψ, in Ω, 2.a x x ψ =, ψ = 0, on y =, 2.b x ψ = 0, ψ = 0, on x = 0. 2.c x The solution of problem 2. can be determined by considering the asymptotic expansion ψr, θ = k= rα k ψk θ; Re, where the exponents α k are allowed to be complex and satisfy Rα < Rα 2 <, and the polar coordinates r, θ satisfy x, y = r cos θ, + r sin θ, π/2 θ 0. The asymptotic expansions of u, v and p can be derived from the asymptotic expansion of ψ, since u and v are related to ψ by definition, and p is related to u and v by the steady incompressible Navier Stokes equations. The leading terms in the asymptotic expansions of ψ, u, v and p are ψr, θ = rψ θ + r 2 ψ 2 θ; Re + r 3 ψ 3 θ; Re + Or α 4, 2.2 ur, θ = u θ + ru 2 θ; Re + r 2 u 3 θ; Re + Or α 4, 2.3 vr, θ = v θ + rv 2 θ; Re + r 2 v 3 θ; Re + Or α 4, 2.4 pr, θ = r p θ; Re + log r p 2 θ + rp 3 θ; Re + Or α 4 2, 2.5 where the functions ψ k, u k, v k and p k, k =, 2, 3 are listed in Appendix A. The formulae for u A, v A and p A are derived by taking as many as three of the leading terms in equations

6 2.3, 2.4 and 2.5 respectively. The regularity of ū, v and p increase with the number of terms in u A, v A and p A respectively. The formulae for u D, v D and p D are derived in a similar manner, that is by considering the steady-state streamfunction ψ, which satisfies ψ 4 2 ψ ψ = Re x ψ =, ψ x ψ = 0, ψ x ψ x 2 ψ, in Ω, 2.6a = 0, on y =, 2.6b = 0, on x =. 2.6c The solution of problem 2.6 can be determined by considering an asymptotic expansion of ψ = ψr, θ analogous to asymptotic expansion 2.2, where now the polar coordinates r, θ satisfy x, y = + r cosθ, + r sin θ, π/2 θ 3π/2. The leading terms in the asymptotic expansions of ψ, u, v and p are ψr, θ = rψ π θ r 2 ψ 2 π θ; Re + r 3 ψ 3 π θ; Re + Or α 4, 2.7 ur, θ = u π θ ru 2 π θ; Re + r 2 u 3 π θ; Re + Or α 4, 2.8 vr, θ = v π θ + rv 2 π θ; Re r 2 v 3 π θ; Re + Or α 4, 2.9 pr, θ = r p π θ; Re + log r p 2 π θ rp 3 π θ; Re + Or α 4 2, 2.0 where the functions ψ k, u k, v k and p k, k =, 2, 3 are equivalent to those expressed in equations 2.2, 2.3, 2.4 and 2.5. The formulae for u D, v D and p D are derived by taking as many as three of the leading terms in equations 2.8, 2.9 and 2.0 respectively, depending on the regularity desired in ū, v and p. A side remark is that α 4, which takes the approximate value of 3.74 +.3i, is an eigenvalue of the Stokes problems, or more precisely problems 2. and 2.6 with Re = 0. The r α 4 terms in equations 2.2 and 2.7 are corresponding eigenfunctions of the Stokes problems, but the eigenfunctions cannot be determined locally [4]. For convenience, the notations ũ = u A + u D, ṽ = v A + v D and p = p A + p D are introduced, and the solutions u A, v A, p A and u D, v D, p D are written in the form u A = K u A k r, θ; Re, va = K vk A r, θ; Re, pa = K p A k r, θ; Re, 2. u D = k= K u D k r, θ; Re, v D = k= K vk D r, θ; Re, p D = k= K p D k r, θ; Re, 2.2 k= k= k= where K {, 2, 3}, and u A k, va k, pa k, ud k, vd k and pd k refer to the kth leading terms in equations 2.3, 2.4, 2.5, 2.8, 2.9 and 2.0 respectively. Writing u = ũ + ū, v = ṽ + v and p = p + p in the unsteady incompressible Navier

7 Stokes equations, and recalling identities B., B.2 and B.3, yield ū t + ũ+ū ũ+ū + ṽ+ v ũ+ū x v t + ũ+ū ṽ+ v + ṽ+ v ṽ+ v x ū x + v N ũ, ṽ + p x = Re 2 ū, in Ω, 2.3 N 2 ũ, ṽ + p = Re 2 v, in Ω, 2.4 = 0, in Ω, 2.5 where 0, if K =, u P u P N ũ, ṽ = x + u P vp, if K = 2, P=A,D u P u P 2 x + u P vp 2 + u P up 2 x + u P vp 2, if K = 3, P=A,D 0, if K =, u P v P N 2 ũ, ṽ = x + v vp P, if K = 2, P=A,D u P v2 P x + v vp 2 P + v up P 2 x + v vp P 2, if K = 3. P=A,D 2.6 2.7 The initial condition reduces to ū, v = ũ, ṽ. The boundary conditions reduce to ū, v = ũ, ṽ on the wall y =, and ū, v = ũ, ṽ on the other three walls. For future reference, K is called the regularity parameter. Gauge Formulation Equations 2.3, 2.4 and 2.5 are just a form of the unsteady incompressible Navier Stokes equations, in which ū, v is the velocity, p is the pressure, and nonlinear terms involving ũ and ṽ are forcing terms. Writing ū, v = a+ φ/ x, b+ φ/, and imposing p = φ/ t + Re 2 φ, yield + ũ + ū ũ + ū x ũ + ū + ṽ + v a t b + v + ũ + ū ṽ + ṽ + v t x ṽ + v N ũ, ṽ = Re 2 a, in Ω, 2.8 N 2 ũ, ṽ = Re 2 b, in Ω, 2.9 2 φ = a x b, in Ω. 2.20 The solution of the lid-driven cavity flow is, therefore, given by the solution of equations 2.8, 2.9 and 2.20, subject to appropriate initial and boundary conditions

8 for a, b and φ. The initial conditions are a = ũ, b = ṽ and φ = 0. The boundary conditions corresponding to a Dirichlet gauge formulation are φ = 0, a = ũ, b = ṽ φ, on y =, 2.2a φ = 0, a = ũ, b = ṽ φ, on y = 0, 2.2b φ = 0, a = ũ φ, x b = ṽ, on x =, 2.2c φ = 0, a = ũ φ, b = ṽ, on x = 0. 2.2d x The boundary conditions corresponding to a Neumann gauge formulation are φ φ = 0, a = φ = 0, a = ũ, x φ = 0, a = ũ, x 3 Numerical Method φ = 0, a = ũ, b = ṽ, on y =, 2.22a x Chebyshev Spectral Collocation Method ũ φ, b = ṽ, on y = 0, 2.22b x φ b = ṽ, on x =, 2.22c φ b = ṽ, on x = 0. 2.22d The fundamental idea of spectral collocation methods for solving ordinary and partial differential equations, is to approximate solutions using interpolants, which satisfy the equations exactly at selected grid points. Chebyshev spectral collocation methods form a class of spectral collocation methods, in which the interpolants are algebraic polynomials, and the grid points are nodes of Chebyshev Gauss Lobatto quadrature formulae. The canonical domain for the application of a Chebyshev spectral collocation method in one-dimension, is the interval [, ]. The grid points are defined by x i = cosπi/n, where i = 0,,..., N, for some positive integer N. The interpolant of the grid data f i fx i for some unknown function f, is the unique Lagrange interpolating polynomial of degree N, given by N N k=0,k j gx = x x k N k=0,k j x j x k f j fx. 3. j=0 The first derivative of the interpolant evaluated at the grid points, can be expressed in terms of a matrix-vector product, such that N g x i = D N ij f j D N f i f x i, 3.2 j=0

9 where f i = f i, and D N is the first-order Chebyshev differentiation matrix. Formulae for the entries of D N can be found in the spectral methods literature [5, 6,9,20,2], and are given by 2N 2 +, if i = j = 0, 6 2N2 +, if i = j = N, D N ij = x 6 i if i = j and i 0, N, 2x 2 i, c i i+j, if i j, c j x i x j 3.3 where c 0 = c N = 2, c = c 2 = = c N =, and i, j = 0,,..., N. The result of equation 3.2 can be readily extended to higher derivatives, where the q th -order Chebyshev differentiation matrix D q N, is given by D N q. The construction of Chebyshev differentiation matrices using formula 3.3 and taking matrix products, is computationally expensive and susceptible to rounding errors. Baltensperger and Trummer [] suggest computing the off-diagonal entries recursively, and the diagonal entries using a negative sum trick, so that q [ i c i D q D q N x i x j j N ii D q N ij ], if i j, c j ij = N D q N ik, if i = j, k=0,k i 3.4 where q is any positive integer, D 0 N is the identity matrix, and i, j = 0,,..., N. The evaluation of Lagrange interpolants using formula 3., is computationally expensive and numerically unstable. Berrut and Trefethen [2] suggest using the barycentric Lagrange interpolation formula, so that f i, if x = x i, N j f j gx = j=0 c j x x j 3.5 N j, if x x i. j=0 c j x x j where i = 0,,..., N. The results presented so far can be readily extended to multiple dimensions. The canonical domain for the application of a Chebyshev spectral collocation method in two-dimensions, is the square [, ] [, ]. The grid points are defined by x i, y j = cosπi/n x, cosπj/n y, where i = 0,,..., N x and j = 0,,..., N y, for some positive integers N x and N y. The interpolant of the grid data f ij fx i, y j for some unknown function f, is the two-dimensional analogue of 3..

0 If a matrix F is defined such that F ij = f ij, then the partial derivatives of the interpolant evaluated at the grid points, can be expressed in terms of matrix-matrix products, where differentiation with respect to x corresponds to premultiplying the columns of F by D Nx, and differentiation with respect to y corresponds to postmultiplying the rows of F by D T N y. This last result provides a straightforward way of spatially discretizing the Dirichlet and Neumann gauge formulations. Exponential Time Differencing Method The most common time stepping methods for solving time-dependent partial differential equations, are low-order finite difference methods, or variants such as implicit-explicit, predictor-corrector and Runge-Kutta methods. Other time stepping methods known for solving stiff nonlinear time-dependent partial differential equations, are integrating factor, split step and slider methods. Exponential time differencing methods, introduced by Cox and Matthews [8], have been shown to outperform the aforementioned methods in terms of accuracy and stability [6]. The fundamental idea of exponential time differencing methods for solving time-dependent partial differential equations, is to solve the linear part of the equations exactly, and to approximate the nonlinear part of the equations using methods of choice. As an illustration, consider the system of equations obtained from the spatial discretization of a time-dependent partial differential equation, such that df dt = cf + Nf, t, 3.6 where c is a known scalar, N is a vector composed of forcing terms and nonlinear terms, and f is a vector of unknowns. Multiplying equation 3.6 by the integrating factor e ct, and integrating from time t = t n to t = t n+, yield ft n+ = e c t ft n + e ctn+ t tn+ t n e cτ Nfτ, τ dτ, 3.7 where t n = n t, and n = 0,, 2,.... Different exponential time differencing schemes are derived by applying different approximations to the integral in equation 3.7 [8]. The first-order exponential time differencing scheme is derived by assuming that N is constant in time, so that f n+ = e c t f n + ec t Nf n, t n, 3.8 c where f n denotes the numerical approximation of ft n. The direct evaluation of formula 3.8 is numerically unstable, due to cancellation errors arising from the inaccurate computation of e c t /c, for small c. Kassam and Trefethen [6] resolved similar difficulties in the fourth-order exponential time differencing Runge-Kutta scheme, by using complex contour integrals, so that e c t c = 2πi Γ e z t z z c dz, 3.9

where Γ is any closed contour in the complex plane that encloses c. Choosing Γ to be the unit circle centred at c, and approximating the resulting integral using the periodic trapezoid rule, yield e c t c N z N z k= e z k t z k, 3.0 where z k = c + e i2πk/nz, and N z is a positive integer, typically less than 00. In general, f satisfies the system of equations df/dt = Lf + Nf, t, where L is a nondiagonal matrix. The application of an exponential time differencing method to this system of equations, involves the computationally expensive tasks of calculating inverses and exponentials of matrices. This is undesirable for problems in multiple dimensions, where sizes of matrices are typically very large. In two-dimensions, f may be replaced by a matrix of unknowns F, as with twodimensional Chebyshev spectral collocation methods. The resulting system of equations may consist of matrices premultiplying and postmultiplying F, and so cannot be discretized using a direct exponential time differencing method, unless the system of equations is transformed, by some suitable diagonalization, into a system of equations resembling that given in equation 3.6. This last result provides a way of temporally discretizing the Dirichlet and Neumann gauge formulations. Discretization Grid points x i, y j are defined such that x i = + cosπi/n x /2 and y j = + cosπj/n y /2, where i = 0,,..., N x and j = 0,,..., N y, for some positive integers N x and N y. Discrete time t n is defined such that t n = n t, where n = 0,, 2,... and t is some time step. Matrices A, B, P, U and V are defined such that A n ij, Bn ij, Pn ij, Un ij and V n ij respectively denote numerical approximations of a, b, φ, ū and v evaluated at x, y, t = x i, y j, t n. Differentiation matrices D x = 2D Nx, D y = 2DN T y, D xx = 4D 2 N x are defined such that D x A n ij, AD y n ij, D xxa n ij and AD yy n ij and D yy = 4D 2 T N y respectively denote numerical approximations of a/ x, a/, 2 a/ x 2 and 2 a/ 2 evaluated at x, y, t = x i, y j, t n, and similarly for the spatial derivatives of b, φ, ū and v. The problems of finding a, b, φ, ū and v from the Dirichlet and Neumann gauge formulations defined in Section 2 are, therefore, reduced to the problems of finding A, B, P, U and V from the resulting systems of discrete equations. Computing A & B A and B satisfy discrete heat equations, subject to discrete and temporally explicit Dirichlet boundary conditions, such that the computations of A, B and P are decoupled. Matrix diagonalization procedures, introduced by Lynch, Rice and Thomas [7] for finite difference methods, but later considered by Ehrenstein and Peyret [3] for Chebyshev

2 spectral collocation methods, can be used to compute A and B directly and efficiently, as will be described below. Suppose that a matrix of unknowns M satisfies the discrete heat equation dm ij dt = αd xx M ij + βmd yy ij + L ij, for i I, j J, 3. where I = {, 2,..., N x }, J = {, 2,..., N y }, α, β are given scalars, and L is a given matrix. Suppose that M also satisfies discrete Dirichlet boundary conditions, so that formulae for the boundary entries of M are given. Substituting these formulae into equation 3., yields a matrix equation of the form d M dt = D xx M + M Dyy + L, 3.2 where M is the interior of M. Assuming that D xx and D yy are diagonalizable, there exist diagonal matrices Λ x and Λ y whose entries are eigenvalues of D xx and D yy respectively, and invertible matrices X and Y whose columns are eigenvectors of Dxx and D yy respectively, for which D xx = XΛ x X and D yy = Y Λ y Y. Premultiplying and postmultiplying equation 3.2 by X and Y respectively, yield d ˆM dt = Λ x ˆM + ˆMΛy + X LY, 3.3 where ˆM = X MY. Recalling the diagonality of Λ x and Λ y, yields d ˆM ij dt = λ ij ˆM ij + X LY ij, for i I, j J, 3.4 where λ ij = Λ x ii + Λ y jj. Equation 3.4 resembles equation 3.6, for which exponential time differencing methods are applicable. The first-order exponential time differencing scheme is sufficient for computing steady-state solutions. Applying formula 3.8 to equation 3.4, yields ˆM n+ ij = e λ ij t ˆM n ij + eλ ij t λ ij X LY n ij, for i I, j J, 3.5 where the coeffecients e λ ij t /λ ij are calculated using formula 3.0. The interior entries of M are the entries of M = X ˆMY, whilst the boundary entries of M are given by the discrete Dirichlet boundary conditions, and so the problem is solved, provided that assumptions made about the diagonalizability of D xx and D yy are justified. Gottlieb and Lustman [4] proved that matrices similar to D xx and D yy, possess real, negative and distinct eigenvalues, and hence are diagonalizable. Since A and B satisfy equations and boundary conditions similar to M, the above algorithm can be used to compute A and B.

3 Computing P P satisfies a discrete Poisson equation, subject to discrete Dirichlet or discrete Neumann boundary conditions, depending on whether a Dirichlet or a Neumann gauge formulation is used. A matrix diagonalization procedure, can be used to compute P directly and efficiently, as will be described below. Suppose that a matrix of unknowns M satisfies the discrete Poisson equation D xx M ij + MD yy ij = L ij, for i I, j J, 3.6 and the discrete Robin boundary conditions αm ij + βmd y ij = 0, for i I, j = 0, N y, 3.7a αm ij + βd x M ij = 0, for i = 0, N x, j J, 3.7b where I = {, 2,..., N x }, J = {, 2,..., N y }, α, β are given scalars, and L is a given matrix. Formulae for the boundary entries of M can be derived from boundary condition 3.7. Substituting these formulae into equation 3.6, yields a matrix equation of the form D xx M + M Dyy = L, 3.8 where M is the interior of M. Assuming that D xx and D yy are diagonalizable, there exist diagonal matrices Λ x and Λ y whose entries are eigenvalues of D xx and D yy respectively, and invertible matrices X and Y whose columns are eigenvectors of Dxx and D yy respectively, for which D xx = XΛ x X and D yy = Y Λ y Y. Premultiplying and postmultiplying equation 3.8 by X and Y respectively, yield Λ x ˆM + ˆMΛy = X LY, 3.9 where ˆM = X MY. Recalling the diagonality of Λx and Λ y, yields ˆM ij = λ ij X LY ij, for i I, j J. 3.20 where λ ij = Λ x ii + Λ y jj are assumed to be nonzero. The interior entries of M are the entries of M = X ˆMY, whilst the boundary entries of M can be recovered from the discrete Robin boundary conditions, and so the problem is solved, provided that assumptions made about the diagonalizability of D xx and D yy, and the nonzeroness of λ ij in formula 3.20, are justified. If β = 0 so that discrete Dirichlet boundary conditions are prescribed, then D xx and D yy each possess real, negative and distinct eigenvalues [4], and so all assumptions are justified. If α = 0 so that discrete Neumann boundary conditions are prescribed, then D xx and D yy each possess one zero eigenvalue, in addition to other real, negative and distinct eigenvalues [4], and so all assumptions are justified, apart from λ ij = 0 occuring once in formula 3.20. The corresponding ˆM ij may, nevertheless, be set equal to zero [20], in consistency with the fact that a Poisson equation, subject to Neumann boundary conditions, is solvable only up to an additive constant. Since P satisfies equations and boundary conditions similar to M, the above algorithm can be used to compute P.

4 4 Numerical Results The numerical method described in Section 3, is applied to the Dirichlet and Neumann gauge formulations defined in Section 2, for the flow at Re = 000. The numerical results are compared to benchmark results, which were set by Botella and Peyret [3] using a subtraction method to reduce the pollution effect of singularities at the top corners of the cavity, a projection method to decouple the computation of velocity and pressure, a Chebyshev spectral collocation method to discretize spatially, and a finite difference type method to discretize temporally. For ease of reference, the various schemes considered in this section are denoted by DG K : schemes based on a Dirichlet gauge formulation, NG K : schemes based on a Neumann gauge formulation, NS K : schemes derived by Botella and Peyret [4], where the subscripts K {0,, 2, 3} indicate the regularity parameter used, with K = 0 indicating that a subtraction method is not applied. The benchmark results were set using an NS 2 scheme. An important remark is that the benchmark results correspond to a lid-driven cavity flow, whose lid was set to move in the opposite direction to that considered in this paper, and so the benchmark results displayed in this section are, in fact, adaptations of the benchmark results given in [3]. The DG K and NG K schemes are implemented with N x = N y = N for various values of N, and t = 0.004. The stopping criterion for steady-state solutions is max i,j=0,,...,n { Un+ ij U n ij, V n+ ij V n ij } t max i,j=0,,...,n { Un+ ij, V n+ ij } < 0 0, n = 0,, 2,.... A comparison between the DG 2, NG 2 and NS 2 schemes is made, by calculating the extrema of u and v along the centerlines of the cavity. The calculations are made by interpolating the steady-state approximations of u and v onto an equispaced grid of spacing 0 4 in both coordinate directions. The minimum of u on x = 0.5 is denoted by u min, and the minimum and maximum of v on y = 0.5 are denoted by v min and v max respectively. The locations at which u min, v min and v max are attained, are denoted by y min, x min and x max respectively. Table displays the values of u min, y min, v min, x min, v max and x max, obtained from the DG 2, NG 2 and NS 2 schemes, for increasing values of N. The NG 2 results show good agreement with the NS 2 results for large N, but the DG 2 results seem to converge to the NS 2 results at a slower rate. This is surprising, since a spectral method converges at a rate which depends on the regularity of the solution being computed, and there is no reason why the solution computed by the NG 2 scheme should be more regular than the solution computed by the DG 2 scheme. A comparison between the NG 2 and NS 2 schemes is made, by drawing contour plots of ψ, ω and p, using the N = 28 solution of the NG 2 scheme. The pressure p is normalized such that p = 0 at the center of the cavity. The vorticity ω is computed

5 Table : Extrema of the velocity through the centerlines of the cavity, at Re = 000, for various values of N. Scheme N u min y min v min x min v max x max DG 2 48 0.3884400 0.78 0.5268834 0.9092 0.3767694 0.577 DG 2 64 0.3885525 0.77 0.5270509 0.9092 0.376985 0.578 DG 2 96 0.3885680 0.77 0.5270744 0.9092 0.3769422 0.578 DG 2 28 0.3885694 0.77 0.5270765 0.9092 0.3769442 0.578 NG 2 48 0.3885325 0.77 0.5270302 0.9092 0.376903 0.578 NG 2 64 0.3885708 0.77 0.5270770 0.9092 0.3769452 0.578 NG 2 96 0.3885699 0.77 0.527077 0.9092 0.3769448 0.578 NG 2 28 0.3885698 0.77 0.527077 0.9092 0.3769447 0.578 NS 2 48 0.388527 0.77 0.527068 0.9092 0.376899 0.578 NS 2 64 0.3885695 0.77 0.5270763 0.9092 0.3769439 0.578 NS 2 96 0.3885698 0.77 0.527077 0.9092 0.3769447 0.578 NS 2 28 0.3885698 0.77 0.527077 0.9092 0.3769447 0.578 from the definition ω = u/ + v/ x. The streamfunction ψ is computed from the Poisson equation 2 ψ = ω, subject to homogeneous Dirichlet boundary conditions. The plots are made by interpolating the steady-state approximations of ψ, ω and p onto a 24 24 Chebyshev Gauss Lobatto grid, and using MATLAB to draw contour lines at the contour levels given in [3]. Figures, 2 and 3, which display the contour plots obtained from the NG 2 scheme, are indistinguishable from the corresponding contour plots given in [3]. A further comparison between the NG 2 and NS 2 schemes is made, by calculating the values of u, v, p and ω at various points along the centerlines of the cavity, using the N = 60 solution of the NG 2 scheme. The calculations are made by evaluating the steady-state approximations of u, v, p and ω at the points given in [3]. Tables 3 and 4 display the various values of u, v, p and ω obtained from the NG 2 and NS 2 schemes. The NG 2 results agree well with the NS 2 results, with most differences occuring only in the last decimal digits. A comparison between the convergence rates of the DG K, NG K and NS K schemes is made, by calculating the Chebyshev-weighted L 2 error norms u N u ref and p N p ref, where u N, p N denotes the steady-state approximation of u, p for some value of N, and u ref, p ref denotes the steady-state approximation of u, p obtained from the NG 3 scheme with N = 60. The error norms are calculated by using Chebyshev Gauss Lobatto quadrature formulae, with 24 nodes in both coordinate directions. Figures 4 and 5 display plots of the error norms against N on logarithmic scales, for the DG K and NG K schemes respectively. The error norms behave like ON γ, where the corresponding γ are given in Table 2. A side remark is that numerically stable results could not be obtained from the DG 3

6 a b c d g h f e j i hg a = 0.75, c = 9 0 2, e = 0 2, g = 0 5, i = 5.0 0 4, b = 0.00, d= 5 0 2, f = 0 4, h= 0 4, j =.5 0 3. Figure : Contour lines of the streamfunction, at Re = 000. k i j j i h g f e d k a b c g i h j k j k i e h g f f a = 5.0, c = 3.0, e =.0, g = 0.0, i =.0, k= 3.0, b = 4.0, d= 2.0, f = 0.5, h= 0.5, j = 2.0. Figure 2: Contour lines of the vorticity, at Re = 000.

7 f e c d e c a b d e f j i d h g f e d a = 0.30, c = 0.2, e = 0.09, g = 0.05, i = 0.000, b = 0.7, d= 0., f = 0.07, h= 0.02, j = 0.002. Figure 3: Contour lines of the pressure, at Re = 000. Table 2: Rates of convergence of velocity and pressure for the DG K, NG K and NS K [4] schemes. Error norm DG 0 DG DG 2 NG 0 NG NG 2 NG 3 NS 0 NS NS 2 u N u ref. 2.9 3.3.0 3.2 5. 6.9. 3.0 5. p N p ref 0.9..5 3. 5.0. 3.2

8 Table 3: Horizontal velocity, pressure and vorticity at various points along the vertical centerline of the cavity, at Re = 000. y u NS 2 u NG 2 p NS 2 p NG 2 ω NS 2 ω NG 2.0000.0000000.0000000 0.052987 0.052985 4.7534 4.7537 0.9766 0.6644227 0.6644227 0.052009 0.052007 2.0670 2.0670 0.9688 0.5808359 0.5808359 0.0554 0.055 9.49496 9.49496 0.9609 0.569277 0.569278 0.050949 0.050946 6.95968 6.95968 0.953 0.4723329 0.4723329 0.050329 0.050327 4.85754 4.85755 0.856 0.337222 0.337223 0.03490 0.034908.76200.76200 0.7344 0.886747 0.886747 0.0222 0.0220 2.092 2.092 0.672 0.057078 0.057078 0.000827 0.000828 2.06539 2.06539 0.5000 0.062056 0.0620562 0.000000 0.000000 2.06722 2.06722 0.453 0.08999 0.082000 0.004434 0.004434 2.0625 2.0626 0.283 0.2803696 0.2803697 0.040377 0.040378 2.26772 2.26772 0.79 0.388569 0.388569 0.08925 0.08927.05467.05467 0.06 0.300456 0.300456 0.0487 0.0489.63436.63436 0.0703 0.2228955 0.2228955 0.08566 0.08568 2.2075 2.2075 0.0625 0.2023300 0.2023300 0.09200 0.09202 2.3786 2.3786 0.0547 0.8288 0.82882 0.09689 0.0969 2.44960 2.4496 0.0000 0.0000000 0.0000000 0.059 0.0594 4.6648 4.6669 0 0 0 2 un uref 0 3 pn pref 0 0 2 0 4 0 3 0 5 0 4 0 6 DG 0 0 5 DG DG 2 0 7 32 48 64 8096 28 0 6 32 48 64 8096 28 N N Figure 4: Error norms of velocity and pressure for the DG K schemes, plotted against N on logarithmic scales.

9 Table 4: Vertical velocity, pressure and vorticity at various points along the horizontal centerline of the cavity, at Re = 000. x v NS 2 v NG 2 p NS 2 p NG 2 ω NS 2 ω NG 2.0000 0.0000000 0.0000000 0.077455 0.077454 5.4627 5.4632 0.9688 0.2279225 0.2279226 0.078837 0.078836 8.44350 8.44350 0.9609 0.2936869 0.2936870 0.078685 0.078685 8.2466 8.2466 0.953 0.355323 0.355324 0.07848 0.07848 7.58524 7.58524 0.9453 0.403754 0.403755 0.07754 0.07753 6.50867 6.50866 0.9063 0.5264392 0.5264392 0.06586 0.06585 0.9229 0.9229 0.8594 0.4264545 0.4264545 0.049029 0.049029 3.4306 3.4306 0.8047 0.320237 0.320238 0.034552 0.03455 2.27 2.27 0.5000 0.0257995 0.0257995 0.000000 0.000000 2.06722 2.06722 0.2344 0.3253592 0.3253593 0.044848 0.044849 2.0622 2.0622 0.2266 0.3339924 0.3339924 0.047260 0.047260 2.0074 2.0074 0.563 0.376989 0.376989 0.0695 0.06952 0.74207 0.74207 0.0938 0.3330442 0.3330442 0.084386 0.084386 0.82398 0.82398 0.078 0.3099097 0.3099097 0.08676 0.08677.2399.2399 0.0703 0.2962703 0.2962703 0.087653 0.087653.50306.50306 0.0625 0.2807056 0.2807056 0.088445 0.088445.83308.83308 0.0000 0.0000000 0.0000000 0.090477 0.090477 7.66369 7.66274 0 0 0 2 un uref 0 3 pn pref 0 0 2 0 4 0 3 0 5 0 4 0 6 NG 0 NG 0 5 NG 2 NG 3 0 7 32 48 64 8096 28 0 6 32 48 64 8096 28 N N Figure 5: Error norms of velocity and pressure for the NG K schemes, plotted against N on logarithmic scales.

20 scheme, for the choice of parameters considered in this paper. Nevertheless, a similar convergence analysis for the flow at Re = 0, suggests that the DG 3 scheme yields similar convergence rates to the DG and DG 2 schemes. In comparison with the NG K and NS K schemes, the DG K schemes yield reduced convergence rates for both u and p, indicating that the numerical approximations of φ failed to converge to the exact φ in the Dirichlet gauge formulations. The error estimates derived by Wang and Liu [22], and Nochetto and Pyo [9], for the backward Euler time discretization of Dirichlet gauge formulations, suggest a similar phenomenon of reduced convergence rates. In contrast, the NG K and NS K schemes yield similar convergence rates and, in general, similar numerical results, indicating the applicability of Neumann gauge formulations for incompressible flow calculations. 5 Conclusion The spectral gauge method, which is a spectral method for solving the unsteady incompressible Navier Stokes equations in gauge formulation, was presented. The method yields accurate results if Neumann boundary conditions are imposed on the gauge variable, but suffers from reduced convergence rates if Dirichlet boundary conditions are imposed on the gauge variable. The failure to satisfy a compatability condition similar to that derived by Nochetto and Pyo [9], may be a reason for the reduced convergence rates, but a theoretical study is needed to verify this claim. The method is straightforward to implement, with efficient computation of the solutions made possible by matrix diagonalization procedures. The method can be modified to accommodate different classes of spectral methods, as well as different exponential time differencing schemes. The method can also be extended to include threedimensional incompressible viscous fluid flow problems. References [] R. Baltensperger and M. R. Trummer, 2003. Spectral differencing with a twist. SIAM J. Sci. Comput., 24: 465 487. [2] J. P. Berrut and L. N. Trefethen. Barycentric Lagrange interpolation. SIAM Rev. to appear. [3] O. Botella and R. Peyret, 998. Benchmark spectral results on the lid-driven cavity flow. Comput. & Fluids, 27: 42 433. [4] O. Botella and R. Peyret, 200. Computing singular solutions of the Navier Stokes equations with the Chebyshev-collocation method. Internat. J. Numer. Methods Fluids, 36: 25 63. [5] J. P. Boyd, 200. Chebyshev and Fourier spectral methods. Dover.

2 [6] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, 988. Spectral methods in fluid dynamics. Springer-Verlag. [7] A. J. Chorin, 968. Numerical solution of the Navier Stokes equations. Math. Comp., 22: 745 762. [8] S. M. Cox and P. C. Matthews, 2002. Exponential time differencing for stiff systems. J. Comput. Phys., 76: 430 455. [9] M. O. Deville, P. F. Fischer and E. H. Mund, 2002. High-order methods for incompressible fluid flow. Cambridge University Press. [0] Weinan E and J. G. Liu, 995. Projection method I: convergence and numerical boundary layers. SIAM J. Numer. Anal., 32: 07 057. [] Weinan E and J. G. Liu, 2000. Gauge finite element method for incompressible flows. Internat. J. Numer. Methods Fluids, 34: 70 70. [2] Weinan E and J. G. Liu, 2003. Gauge method for viscous incompressible flow. Commun. Math. Sci., : 37 332. [3] U. Ehrenstein and R. Peyret, 989. A Chebyshev spectral collocation method for the Navier Stokes equations with application to double-diffusive convection. Internat. J. Numer. Methods Fluids, 9: 427 452. [4] D. Gottlieb and L. Lustman, 983. The spectrum of the Chebyshev collocation operator for the heat equation. SIAM J. Numer. Anal., 20: 909 92. [5] M. M. Gupta, R. P. Manohar and B. Noble, 98. Nature of viscous flows near sharp corners. Comput. & Fluids, 9: 379 388. [6] A. K. Kassam and L. N. Trefethen. Fourth-order time stepping for stiff PDEs. SIAM J. Sci. Comput. to appear. [7] R. E. Lynch, J. R. Rice and D. H. Thomas, 964. Direct solution of partial differential equations by tensor product matrices. Numer. Math., 6: 85 99. [8] H. K. Moffatt, 964. Viscous and resistive eddies near a sharp corner. J. Fluid Mech., 8: 8. [9] R. H. Nochetto and J. H. Pyo. Error estimates for semi-discrete gauge methods for the Navier Stokes equations. Math. Comp. to appear. [20] R. Peyret, 2002. Spectral methods for incompressible viscous flow. Springer-Verlag. [2] L. N. Trefethen, 2000. Spectral methods in MATLAB. SIAM. [22] C. Wang and J. G. Liu, 2000. Convergence of gauge method for incompressible flow. Math. Comp., 69: 385 407.

22 Appendix A Formulae for the functions ψ k, u k, v k and p k expressed in equations 2.2, 2.3, 2.4 and 2.5, and the derivatives u k and v k, where k =, 2, 3, are given below. Analogous results for a lid-driven cavity flow, whose lid moves with velocity u, v =, 0, were derived by Gupta, Manohar and Noble [5], and used by Botella and Peyret [3]. ] ψ θ = α [A 0 + A θ cosθ + B 0 + B θ sin θ, ] u θ = α [ A 2 + B 0 + B θ + A 2 cos 2θ + B 2 sin 2θ, ] v θ = α [ B 2 A 0 A θ B 2 cos 2θ + A 2 sin 2θ, [ ] u θ = α B + B cos 2θ A sin 2θ, [ ] v θ = α A + A cos 2θ + B sin 2θ, [ ] p θ; Re = Re α 2 cosθ + π sin θ, ] ψ 2 θ; Re = Re α [C 2 0 + C θ + D 0 + D θ+ D 2 θ 2 cos 2θ + E 0 + E θ+ E 2 θ 2 sin 2θ, [ u 2 θ; Re = Re α 2 C + D 2 + 2E 0 + D 2 + 2E θ + 2E 2 θ 2 cosθ + 2C 0 2D 0 + 2 E + 2C 2D + E 2 θ + 2D 2 θ 2 sin θ ] + D 2 + D 2 θ cos 3θ + E 2 + E 2 θ sin 3θ, [ v 2 θ; Re = Re α 2 C D 2 2E 0 + D 2 2E θ + 2E 2 θ 2 sin θ 2C 0 2D 0 + 2 E + 2C 2D + E 2 θ + 2D 2 θ 2 + cosθ ] + D 2 + D 2 θ sin 3θ + E 2 E 2 θ cos 3θ, [ u 2θ; Re = Re α 2 C 5D 2 2E 0 + E 2 + 5D 2 2E θ + 2E 2 θ 2 sin θ 2C 0 2D 0 + D 2 + 52 E + 2C 2D + 5E 2 θ + 2D 2 θ 2 + cosθ ] + D 2 + 3E 2 + 3E 2 θ cos 3θ + E 2 3D 2 3D 2 θ sin 3θ, [ v 2 θ; Re = Re α 2 C 5D 2 2E 0 + E 2 + 5D 2 2E θ + 2E 2 θ 2 cosθ 2C 0 + 2D 0 D 2 52 E + 2C + 2D 5E 2 θ + 2D 2 θ 2 + + D 2 + 3 E 2 + 3E 2 θ sin 3θ + [ π p 2 θ = α 2 4 + 8π 2 ], 64 ] E 2 + 3 D 2 + 3D 2 θ cos 3θ, sin θ

23 [ ψ 3 θ; Re = Re 2 α 3 F 0 + F θ + F 2 θ 2 + F 3 θ 3 cos θ + G 0 + G θ + G 2 θ 2 + G 3 θ 3 sin θ ] + H 0 + H θ + H 2 θ 2 + H 3 θ 3 cos 3θ + I 0 + I θ + I 2 θ 2 + I 3 θ 3 sin 3θ, [ u 3 θ; Re = Re 2 α 3 F 2 + 2G 0 + F 2 + 2G θ + 3F 2 3 + 2G 2 θ 2 + 2G 3 θ 3 + F 2 G 0 + H 2 + 3I 0 + F 2 G + H 2 + 3I θ + 3F 2 3 G 2 + 3H 2 3 + 3I 2 θ 2 + G 3 + 3I 3 θ 3 cos 2θ + G 2 + F 0 + I 2 3H 0 + G 2 + F + I 2 3H θ + 3 2 G 3 + F 2 + 3 2 I 3 3H 2 θ 2 + F 3 3H 3 θ 3 sin 2θ + H 2 + H 2 θ + 3H 2 3θ 2 cos 4θ ] + I 2 + I 2 θ + 3 I 2 3 θ 2 sin 4θ, [ v 3 θ; Re = Re 2 α 3 G 2 2F 0 + G 2 2F θ + 3G 2 3 2F 2 θ 2 + 2F 3 θ 3 + F 2 G 0 H 2 3I 0 + F 2 G H 2 3I θ + 3F 2 3 G 2 3H 2 3 3I 2 θ 2 + G 3 3I 3 θ 3 sin 2θ G 2 + F 0 I 2 + 3H 0 + G 2 + F I 2 + 3H θ + 3 2 G 3 + F 2 3 2 I 3 + 3H 2 θ 2 + F 3 + 3H 3 θ 3 cos 2θ + H 2 + H 2 θ + 3H 2 3θ 2 sin 4θ ] I 2 + I 2 θ + 3 I 2 3 θ 2 cos 4θ, [ u 3 θ; Re = Re2 α 3 F 2 + 2G + 3F 3 + 4G 2 θ + 6G 3 θ 2 + 2F 0 + F 2 6H 0 + H 2 + 4I + 2F + 3F 3 6H + 3H 3 + 8I 2 θ + 2F 2 6H 2 + 2I 3 θ 2 + 2F 3 6H 3 θ 3 cos 2θ + 2G 0 + G 2 6I 0 + I 2 4H + 2G + 3G 3 6I + 3I 3 8H 2 θ + 2G 2 6I 2 2H 3 θ 2 + 2G 3 6I 3 θ 3 sin 2θ + H 2 + 2I + 3H 3 + 4I 2 θ + 6I 3 θ 2 cos 4θ ] + I 2 2H + 3I 3 4H 2 θ + 6H 3 θ 2 sin 4θ,

24 [ v 3θ; Re = Re 2 α 3 G 2 2F + 3G 3 4F 2 θ + 6F 3 θ 2 + 2F 0 + F 2 + 6H 0 H 2 4I + 2F + 3F 3 + 6H 3H 3 8I 2 θ + 2F 2 + 6H 2 2I 3 θ 2 + 2F 3 + 6H 3 θ 3 sin 2θ 2G 0 + G 2 + 6I 0 I 2 + 4H + 2G + 3G 3 + 6I 3I 3 + 8H 2 θ p 3 θ; Re = Re α 3 [ 33π 6 + 552π 4 6264π 2 + 9632 where + 2G 2 + 6I 2 + 2H 3 θ 2 + 2G 3 + 6I 3 θ 3 cos 2θ + H 2 + 2I + 3H 3 + 4I 2 θ + 6I 3 θ 2 sin 4θ ] I 2 2H + 3I 3 4H 2 θ + 6H 3 θ 2 cos 4θ, + 3π3 28π θ + π2 20 36864 256 28 9π 7 + 24π 5 24π 3 + 752π + + π6 0π 4 + 72π 2 + 256 θ 36864 024 + π5 3π 3 + 20π θ 2 + π2 4 θ 3 sin θ 256 96 π 6 22π 4 80π 2 + 352 + 4096 π 5 6π 3 96π + + π4 4π 2 + 24 024 256 α = π2 4, A 0 = 0, A =, B 0 = π2 4, B = π 2, θ 2 + π3 4π θ 3 cosθ 92 + π3 + 36π θ + 3π2 4 256 28 θ2 cos 3θ θ + π3 + 2π ] θ 2 sin 3θ, 256 C 0 = π5 4π 3 + 24π, C = π4 8π 2, 024 256 D 0 = π5 + 4π 3 24π, D = π2 2, 024 28 D 2 = π 6, E 0 = π4 + 0π 2 + 24 52 E = π3 + 6π, E 2 = π2 4 64 64, F 0 = 9π7 32π 5 424π 3 22π 29492, F = π6 0π 4 + 6π 2 + 224, 892,

25 F 2 = 5π3 + 2π 2048 G 0 = 39π6 + 23π 4 3960π 2 80 47456 G 2 = 2π4 + 3π 2 + 2 024, F 3 = π2 4, 384, G = 3π5 + 8π 3 40π, 2048 H 0 = 9π7 + 32π 5 + 424π 3 + 22π 29492 H 2 = π3 56π 644 I 0 = 9π6 85π 4 + 888π 2 80 47456 I 2 = 3π4 + 27π 2 52 3072, G 3 = π3 + 4π, 768, H = 3π6 + 02π 4 + 504π 2 856, 73728, H 3 = 3π2 + 4, 768, I = 9π5 + 07π 3 + 588π, 8432, I 3 = 3π3 36π. 4608 Appendix B The functions u A, v A, p A, u D, v D and p D defined in equations 2. and 2.2, satisfy the following identities p P x = Re 2 u P, p P = Re 2 v P, u P x + vp = 0, B.a B.b B.c u P u P x + vp u P + pp 2 x = Re 2 u P 2, B.2a u P v P x + vp v P + pp 2 = Re 2 v P 2, u P 2 x + vp 2 = 0, B.2b B.2c u P u P 2 x + vp u P 2 + up 2 u P x + vp 2 u P + pp 3 x = Re 2 u P 3, B.3a u P where P = A, D. v P 2 x + vp v P 2 + up 2 v P x + vp 2 u P 3 x + vp 3 = 0, v P + pp 3 = Re 2 v P 3, B.3b B.3c