A robust layer-resolving spline collocation method for a convection–diffusion problem

Abstract

We consider finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. The problem is numerically treated by a quadratic spline collocation method on a piecewise uniform slightly modified Shishkin mesh. The position of collocation points is chosen so that the obtained scheme satisfies the discrete minimum principle. We prove pointwise convergence of order $O(N^{-2}{\ln }^{2}N)$ inside the boundary layer and second order convergence elsewhere. The uniform convergence of the approximate continual solution is also given. Further, we approximate normalized flux and give estimates of the error at the mesh points and between them. The numerical experiments presented in the paper confirm our theoretical results.

Introduction

Differential equations with a small parameter multiplying the highest order derivatives are called singularly perturbed differential equations. They arise in modeling numerous processes in nature, e.g. in fluid and gas dynamics, semi-conductor modeling, aerodynamics, modeling of water and atmospheric pollution, financial modeling, etc. Their solutions are characterized by the presence of boundary layers, i.e. regions where the solution and its derivatives change rapidly. In general, classical numerical methods fail to catch those rapid changes of the solution which makes them inappropriate for singularly perturbed problems. In order to overcome this, one may use the so-called ε-uniform methods in the sense of the following definition, see [5].

Definition 1.1

Let $u_{\epsilon }$ be the solution of a singularly perturbed problem and let $U_{\epsilon }^N$ be a numerical approximation of u obtained by a numerical method with N degrees of freedom. The numerical method is said to be ε-uniformly convergent with the respect to the perturbation parameter $\epsilon$ in the norm $\Vert ·\Vert$ if:

$$\Vert u_{\epsilon }-U_{\epsilon }^N\Vert ⩽\vartheta (N)\text{for}N⩾N_0$$

with a function $\vartheta$ and a threshold value $N_0>0$ that are both independent of $\epsilon$ and

$$\underset{N\to \infty }{\lim }\vartheta (N)=0\text{.}$$

A variety of $\epsilon$-uniformly convergent numerical methods have been proposed and analyzed in the literature, see [4], [5], [7], [8], [9]. Here we focus our attention on the construction of the so-called robust layer-resolving $\epsilon$-uniformly convergent numerical method, since such methods are often needed in real applications. According to Farrell et al. [3], a numerical method is layer-resolving if at each point of the solution domain generates numerical approximations of the exact solution and its derivatives for all values of the singular perturbation parameter from finite to arbitrarily small values. Also, numerical approximations are said to be robust if they inherit the stability properties of the exact solution by preserving the monotonicity of the original problem.

We consider the following singularly perturbed convection–diffusion model problem:

$$\begin{array}{c}\mathcal{L}y≔\epsilon y^{″}+a(x)y^{\prime }-b(x)y=f(x)\text{,}x\in \Omega =(0\text{,}1)\text{,}\hfill \\ y(0)={\gamma }_0\text{,}y(1)={\gamma }_1\text{,}\hfill \end{array}$$

where $0<\epsilon \ll 1$ is a small parameter. The functions a, b and f are assumed to be sufficiently smooth and to satisfy the conditions:

$$a(x)⩾a_{\ast }>0\text{,}b(x)⩾\beta >0\text{for}x\in [0\text{,}1]\text{.}$$

The differential operator $\mathcal{L}$ satisfies the following minimum principle on $\Omega$.

Theorem 1.1 Minimum Principle

Assume that $\phi (0)⩾0$ and $\phi (1)⩾0$, for any function $\phi \in C^2(\overline{\Omega })$. Then $\mathcal{L}\phi (x)⩽0$ for all $x\in \Omega$ implies that $\phi (x)⩾0$ for all $x\in \overline{\Omega }$.

The numerical method constructed in this paper is based on difference scheme which is obtained by

• Quadratic spline function as an approximation function in the collocation process.

• Layer-adapted Shishkin mesh.

The spline function has its first derivative continuous at the layer region, but outside the layer first derivative may have discontinuities. Nonetheless, the obtained method is robust and $\epsilon$-uniformly convergent in the sense of the above definitions.

In [15], the problem (1) is treated by the spline collocation method as well, but only in case $b(x)=0$. The quadratic spline function and the Shishkin mesh are also used for the construction of the numerical method, but the collocation points are placed in the standard way – in the middle between the mesh points. The obtained method is non-monotone and therefore the uniform convergence at the mesh points is proved using Green’s grid function.

In this paper the collocation points are moved in order to obtain the validity of the discrete minimum principle which enables application of the barrier functions technique. The proof of the uniform convergence is much simpler than the proof from Teofanov and Uzelac [15], and the case $b(x)\ne 0$ is involved. Moreover, the more precise estimate of the error outside the boundary layer is obtained.

A family of inverse monotone schemes is examined in [1] for a convection–diffusion problem in conservation form. On the piecewise uniform Shishkin mesh, the error of order:

$$O(N^{-2}{\ln }^{2}N)$$

is proved. The error bound of the same order is derived in [2] for a modified version of Samarskii˘’s monotone scheme on a Shishkin mesh. The combination of central differencing on the fine part of the Shishkin mesh and midpoint upwind approximation on the coarse mesh is the basis for the methods of Stynes and Roos [10] and Stynes and Tobiska [11]. The obtained methods are also inverse monotone and the error of order (2) is proved in the case of $b(x)=0$. In the convergence analysis given in [1], [2], [11] and also in [15], Green’s grid function and $(\Vert ·{\Vert }_{\infty \text{,}d}\text{,}\Vert ·{\Vert }_{1\text{,}d})$ stability result is used. We conjecture that this technique can also be applied in the case of spline difference scheme derived in the present paper. However, here we use $(\Vert ·{\Vert }_{\infty \text{,}d}\text{,}\Vert ·{\Vert }_{\infty \text{,}d})$ stability which is sufficient to obtain the error estimate of order (2).

The main virtue of the collocation method examined in the present paper lies in the possibility of simple and direct constructing of a continuous approximate solution and normalized flux between the mesh points.

The idea of moving collocation points in order to achieve inverse monotonicity is exploited in [14] for the first time for a two parameter singularly perturbed problem. The uniform pointwise convergence is proved in [14] but not for the case considered in the present paper.

A piecewise uniform Shishkin mesh is used as the basis for our discretization because of its simple structure. Performances of Shishkin meshes are in general inferior to those of smoother Bakhvalov meshes. On Bakhvalov meshes the logarithmic factor in the error estimate disappear. Therefore these meshes yield a higher rate convergence, but they are more complicated to construct and analyze.

The paper is organized as follows. In Section 2, we recall the decomposition of the problem (1) and its properties and give the construction of Shishkin mesh and derivation of the spline difference scheme. Section 3 is devoted to the construction of the barrier function for the boundary layer function. In Section 4, we prove the uniform pointwise convergence of the method. Section 5 is devoted to the proof of the uniform pointwise convergence of the normalized flux, while Section 6 contains the proof of uniform convergence of the normalized flux between the mesh points on a slightly modified Shishkin mesh. In Section 7, we give the convergence result for the continual solution. Finally, the numerical results are presented in Section 8.

Throughout the paper C is a generic positive constant independent of the parameter $\epsilon$ and the number of mesh points.