## Abstract

We study the linear stability of some Marangoni flows (thin films) on an inclined plane. The Orr-Sommerfeld eigenproblem contains two non standard boundary conditions of second and third orders. Both depend on the eigenparameter as well as on some non dimensional physical parameters, i.e., Reynolds and capillary numbers and dimensionless surface stress. The basic state is a parabolic profile with the linear part depending on the surface stress. Using a long wave approximation we find a critical value of Reynolds number. The eigenproblem is comparatively solved by a modified Chebyshev-tau approximation. We construct bases in both trial and test spaces such that the differentiation matrices involved are sparse and better conditioned than those involved in the classical tau. Our method is more stable than the classical tau approach and the spurious eigenvalues are completely removed.

## Authors

**Pop**

Babeș-Bolyai University, Faculty of Mathematics and Informatics

**Gheorghiu**

Tiberiu Popoviciu Institute of Numerical Analysis

## Keywords

Chebyshev-Galerkin spectral method; differential eigenvalue problem

### References

See the expanding block below.

## Paper coordinates

I.S. Pop, C.I. Gheorghiu, *A Chebyshev-Galerkin method for fourth order problems*, Proceedings of ICAOR (International Conference on Approximation and Optimization – Romania), Cluj-Napoca, July-August 1996, vol. II, pp. 217-220.

## About this paper

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Approximation and Optimization

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973-98180-7-2

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## Paper (preprint) in HTML form

A Chebyshev-Galerkin method for fourth order problems

## 1. Introduction

Spectral methods have been intersively studied in the last two decades because of their good approximation properties. This advantage was shadowed by some difficulties generated by this discretisation. Thus, the matrices which arise in the spectral discretisation of the differential equations are generally full and have also an increased condition number. Therefore, especially for the fourth order problems, stability and numerical accuracy of the computation can be strongly, affected when a discretization with a large number of elements is used, so the theoretical accuracy of these methods can be lost. There are several works concerned with the problems mentioned above in any of the three existing types of the spectral methods (see, for example [4], [3] for the tau method, [2], [5] for the collocation one or [7] for the Galerkin approach).

In Section 2 an efficient implementation of the Chebyshev-Galerkin method for fourth-order problems is proposed. This approach leads to better conditioned matrices which, in the case of constant coefficients, are also banded. The effectiveness of the method is shown in Section 3 on a model-eigenvalue problem, where optimal convergence order is obtained.

## 2. Chebyshev-Galerkin methods for 4^{th} order problems

In this section we describe the Galerkin variant for the Chebyshev spectral method for ${4}^{th}$ order problems. Only Dirichlet boundary conditions are considered but similar ideas can be applied also in other cases. Here we denote by ${L}_{\omega}^{2}(-1,1)$ the space of functions which are square integrable with respect to the weight function $\omega \left(x\right)={\left(1-{x}^{2}\right)}^{-\frac{1}{2}}.$ By ${T}_{k}\left(x\right)=\mathrm{cos}\left(k\mathrm{arccos}\left(x\right)\right),$ $k\in \mathbb{N}$ we denote the ${k}^{th}$ order Chebyshev polynomials. The following properties are important for the spectral discretization (see, e.g., [1])

${T}_{k+2}\left(x\right)$ | $=2x{T}_{k+1}\left(x\right)-{T}_{k}\left(x\right),$ | $2.1$ | ||

${T}_{k}\left(x\right){T}_{p}\left(x\right)$ | $={\displaystyle \frac{1}{2}}\left({T}_{k+p}+{T}_{\left|k-p\right|}\right),$ | |||

${T}_{k}\left(\pm 1\right)$ | $={\left(\pm 1\right)}^{k},{T}_{k}^{\prime}\left(\pm 1\right)={\left(\pm 1\right)}^{k}{k}^{2},$ | |||

${({T}_{k},{T}_{p})}_{0,\omega}$ | $={\displaystyle \frac{\pi}{2}}{c}_{k}{\delta}_{k,p,}$ |

where $k,p\in \mathbb{N}$, ${\delta}_{k,p}$ is the Kronecker symbol and ${c}_{i}=\{\begin{array}{cc}2,\hfill & \text{if}i=0;\hfill \\ 1,\hfill & \text{if}i0.\hfill \end{array}$

If ${u}^{N}={\sum}_{k=0}^{N}{a}_{k}{T}_{k},$ its ${4}^{th}$ order derivative can be expressed in the following form ([6-Orszag])

$${\left({u}^{N}\right)}^{\left(iv\right)}=\sum _{k=0}^{N}{a}_{k}^{\left(4\right)}{T}_{k}=\sum _{k=0}^{N}{\left({D}^{4}a\right)}_{k}{T}_{k},$$ | $2.2$ |

with ${\left({D}^{4}a\right)}_{k}={\sum}_{j=0}^{N}{d}_{kj}^{4}{a}_{j}$ and

$${c}_{i}{d}_{ij}^{4}=\{\begin{array}{cc}j[{j}^{2}{\left({j}^{2}-4\right)}^{2}-3{i}^{2}{j}^{4}+3{i}^{4}{j}^{2}-{i}^{2}{\left({i}^{2}-4\right)}^{2}],\hfill & \text{if}j=i+4,i+6,\mathrm{\dots},\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}$$ |

The condition number of this matrix is $O\left({N}^{8}\right).$ In the case of homogenous Dirichlet boundary conditions, W. Heinrichs ([3]) proposed the following approach

$${u}^{N}=\sum _{k=0}^{N}{a}_{k}{\mathrm{\Psi}}_{k},\text{where}{\mathrm{\Psi}}_{k}\left(x\right)={\left(1-{x}^{2}\right)}^{2}.{T}_{k}\left(x\right),k=\overline{0,N}$$ | $2.3$ |

This basis satisfies apriori homogenous boundary conditions, but further the Chebyshev coefficients of this expansion were computed and then imposed to be equal with the corresponding ones from the right hand side of the equation. Therefore, this approach is closer to the tau-spectral method.

J. Shen ([7]) considers another expansion within a Galerkin method

${u}^{N}={\sum}_{k=0}^{N}{a}_{k}{\mathrm{\Phi}}_{k},$ where

$${\mathrm{\Phi}}_{k}\left(x\right)={T}_{k}\left(x\right)-\frac{2\left(k+2\right)}{\left(k+3\right)}{T}_{k+2}\left(x\right)+\frac{\left(k+1\right)}{\left(k+3\right)}{T}_{k+4}\left(x\right),k=\overline{0,N}.$$ | $2.4$ |

For both of these bases, an explicit form of the ${4}^{th}$ order differentiation matrix can be given. The condition number is reduced up to $O\left({N}^{4}\right)$ and the differentiation matrices remain upper triangular, but not banded.

In our approach a Galerkin method is considered using the Heinrichs basis $\left(\{{\mathrm{\Psi}}_{k},k=\overline{0,N}\}\right)$ as trial basis and the Shen basis $\left(\{{\mathrm{\Phi}}_{k},k=0,..,N\}\right)$ for the test one. We have the following.

###### Lemma

For ${\mathrm{\Psi}}_{k}$ and ${\mathrm{\Phi}}_{k}$ defined as above we have ${({\mathrm{\Phi}}_{i},{\mathrm{\Psi}}_{j}^{\left(iv\right)})}_{2}=\frac{2}{\pi}{a}_{ij},$ where

$${a}_{ij}=\{\begin{array}{cc}{c}_{i}\left(i+1\right)\left(i+2\right)\left(i+3\right)\left(i+4\right),\hfill & \text{if}j=i,\hfill \\ -2i\left(i+1\right)\left(i+2\right)\left(i+4\right),\hfill & \text{if}j=i+2,\hfill \\ i{\left(i+1\right)}^{2}\left(i+2\right),\hfill & \text{if}j=i+4,\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}$$ | $2.5$ |

This formula and other similar ones for other operators can be obtained by direct computations using the properties of Chebyshev polynomials. As one could observe, the resulting matrix is tridiagonal. More, it is upper triangular. Its condition number is almost halved, comparing it with the one offered by the Shen method, so this discretization is more stable. Even the simplest preconditioner, the diagonal one (denoted by $P$), is effective, generating a condition of order ${N}^{2}.$ These statements are supported by the results in Table 2.1 for the fourth order derivative, but similar values are obtained also for other operators.

Table 2.1. Condition number for the ${D}^{4}$ discretization matrix in (2.5)

$$\begin{array}{ccccc}N& Cond(A)& cond(A)/{N}^{4}& cond(PA)& cond(PA)/{N}^{2}\\ 16& 3.85\cdot {10}^{3}& 0.059& 2.12\cdot {10}^{1}& 0.083\\ 32& 6.35\cdot {10}^{4}& 0.060& 7.67\cdot {10}^{1}& 0.075\\ 64& 1.10\cdot {10}^{6}& 0.065& 2.88\cdot {10}^{2}& 0.070\\ 128& 1.89\cdot {10}^{7}& 0.070& 1.11\cdot {10}^{3}& 0.068\\ 256& 3.19\cdot {10}^{8}& 0.074& 4.35\cdot {10}^{3}& 0.066\\ 512& 5.30\cdot {10}^{9}& 0.077& 1.71\cdot {10}^{4}& 0.066\end{array}$$ |

Table 3.1. First eigenvalue, $\alpha =1.00,$ $R=10000,$ modified pseudospectral method.

$$\begin{array}{cccc}N& Re\left(\lambda \right)& Im\left(\lambda \right)& lo{g}_{10}\left|\lambda -{\lambda}_{ex}\right|\\ \text{13}& \text{0.36841258081}& \text{0.06487680791}& \text{-0.84}\\ \text{18}& \text{0.23709997536}& \text{0.00441789200}& \text{-3.10}\\ \text{23}& \text{0.23563796411}& \text{0.00151181136}& \text{-2.53}\\ \text{33}& \text{0.23748476605}& \text{0.00368618134}& \text{-4.17}\\ \text{38}& \text{0.23752489827}& \text{0.00373265586}& \text{-5.14}\\ \text{43}& \text{0.23752675297}& \text{0.00373921305}& \text{-6.27}\\ \text{48}& \text{0.23752651907}& \text{0.00373967171}& \text{-7.52}\end{array}$$ |

Exact ([6]): ${\lambda}_{ex}=0.23752648882+0.00373967062i.$

## 3. Application

The efficiency of our approach can be illustrated on a model problem, the celebrated Orr-Sommerferl equation.

${\mathrm{\Phi}}^{\left(iv\right)}-2{\alpha}^{2}{\mathrm{\Phi}}^{\prime \prime}+{\alpha}^{4}\mathrm{\Phi}$ | $=i\alpha \mathrm{Re}[\left(U-\lambda \right)\left({\mathrm{\Phi}}^{\prime \prime}-{\alpha}^{2}\mathrm{\Phi}\right)-{U}^{\prime \prime}\mathrm{\Phi}]$ | ||

$\mathrm{\Phi}\left(\pm 1\right)$ | $={\mathrm{\Phi}}^{\prime}\left(\pm 1\right)=0,x\in [-1,1]$ |

The classical tau method generates two spurious eigenvalues in this
case (see, e.g [6]). We have compared the results with those
obtained by W. Huang and D. M. Sloan ([5]) using a modified
pseudospectral method and also with those generated by the classical
tau method. As one can see from the tables below, an improved
convergence is obtained for the Chebyshev-Galerkin method. This
accuarcy is not lost when the approximation order is increased (we
have tested it up to $N=512$), this being a consequence of the
stabilized treatment of the 4^{th} order derivative. No spurious
eigenvalues are obtained. We have achieved an accuracy up to 8
digits. In simpler cases (for example in the computation of the
eigenvalues of the 4^{th} order derivative) in the machine
precision was rapidly achieved.

## Conclusion

We have proposed an approach for the Chebyshev-Galerkin spectral method. This discretization generates sparse matrices which are better conditioned. Applied to differential eigenvalue problems, it does not generate spurious eigenvalues.

## Acknowledgement

The first author expresses his thanks to Prof. W. Jäger for his guidance.

Table 3.2. First eigenvalue, $\alpha =1.00,$ $R=10000,$ Chebyshev-Galerkin method.

$$\begin{array}{cccc}N& Re\left(\lambda \right)& \mathrm{Im}\left(\lambda \right)& lo{g}_{10}\left|\lambda -{\lambda}_{ex}\right|\\ 13& 0.22440097715& 0.01144196988& -1.82\\ 18& 0.23854356657& 0.00081447143& -2.51\\ 23& 0.23750868627& 0.00347148295& -3.57\\ 33& 0.23752593252& 0.00373317977& -5.19\\ 38& 0.23752666392& 0.00373984116& -6.61\\ 43& 0.23752652365& 0.00373967138& -7.49\\ 48& \mathrm{0..23752648754}& 0.00373966968& -8.41\end{array}$$ |

Exact ([6]): ${\lambda}_{ex}=0.23752648882+0.00373967062i.$

## References

- 1 Spectral methods in fluid dynamics, Springer-Verlag, New York/Berlin, (1988). ,
- 2 Some results about pseudospectral approximation of one-dimensional fourth-order problems, Numer. Math. 58 (1990), 1399—418. ,
- 3 A stabilized treatment of the biharmonic operator with spectral methods, SIAM J. Sci. Stat. Comput. 12 (1991), 1162—1172. ,
- 4 Chebyshev matrix operator method for the solution of integrated forms of linear ordinary differential equations, Acta Mech. (to appear).. ,
- 5 The pseudospectral method for solving differential eigenvalue problems, . ,
- 6 Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech. 50 (1971), 689–703. ,
- 7 Efficient spectral-Galerkin method II. Direct solvers of second and fourth order equations by using Chebyshev polynomials, SIAM J. Sci. Stat. Comput. 16 (1995)), 74–87. ,

[1] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A., Zhang, * Spectral methods in fluid dynamics, * Springer-Verlag, New York/Berlin, 1988.

[2] D. Funaro and W. Heinrichs, * Some results about the pseudospectral approximation of one-dimensional fourt-order problems. * Numer. Math. 58, 1990, 1399-418.

[3] W. Heinrichs, * A stabilized treatment of the biharmonic operator with spectral methods, * SIAM J. Sci. Stat. Comput. 12, 1991, 1162-1172.

[4] M. Heigemann, * Chebyshev matrix operator method for the solution of integrated forms of linear ordinary differential equations, * Acta Mech (to appear).

[5] W. Huang and D.M. Sloan, * The pseudospectral method for solving differential eigenvalue problems,*J. Comput. Phys. 111, 1994, 399-409.

[6] S. A. Orszag, * Accurate solution of the Orr-Sommerfeld stability equations, * J. Fluid Mech. 50, 1971, 689-703.

[7] J. Shen, * Efficient spectral-Galerkin method II. Direct solvers of second and fourth order equations by using Chebyshev polynomials, * SIAM J. Sci. Stat. Comput. 16, 1995, 74-8