Enriched Spectral Method for Stiff Convection-Dominated Equations

Abstract

A novel and simple numerical method for stiff convection-dominated problems is studied in presence of boundary or interior layers. A version of the spectral Chevyshev-collocation method enriched with the so-called corrector functions is investigated. The corrector functions here are designed to capture the stiffness of the layers (see the Appendix), and the proposed method does not rely on the adaptive grid points. The extensive numerical results demonstrate that the enriched spectral methods are very accurate with low computational cost.

Appendix A: Elements of the Singular Perturbation Analysis

To treat the boundary and interior layers in a numerical method, we devise a corrector function to capture the sharpness of the layers. The behaviours of the layers much depend on the regularity of the limit solutions $u^0$. More precisely, the discrepancies of the limit solutions $u^0$ at the boundaries or at the interior point respectively produce the boundary and interior layers. To capture their sharpness, zooming near the layers with certain stretched variables we identify leading order differential operators which determine a corrector function. For more details on the singular perturbation analysis, see e.g. [6, 7, 25, 28, 41, 44].

1.1 Example for the Boundary Layer

In the following convection–diffusion equations with constant coefficients

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon \partial _x^2 u^\varepsilon - \partial _x u^\varepsilon = f, \quad x \in (-1,1),\\ u^\varepsilon (-1) = u^\varepsilon (1) = 0, \end{array}\right. } \end{aligned}$$

(A.1)

the formal limit problem, i.e., when $\varepsilon =0$, is

$$\begin{aligned} {\left\{ \begin{array}{ll} -\partial _x u^0 = f,\\ u^0(1) = 0. \end{array}\right. } \end{aligned}$$

(A.2)

A boundary condition is assigned at the inflow $x=1$. Observing that $u^\varepsilon - u^0 \ne 0$ at $x=-1$ and 0 at $x=1$, we find that the discrepancy at the outflow $x=-1$ leads to a sharp transition there. Introducing the stretching variable $\bar{x}$ with $x = \varepsilon \bar{x} - 1$, we identify the leading differential operators

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} - \partial ^2_{\bar{x}} \theta - \partial _{\bar{x}} \theta = 0,\\ \theta = -u(-1), &{}\text {at}\; \bar{x}=0,\\ \theta \rightarrow 0&{} \text {as}\; \bar{x} \rightarrow \infty . \end{array}\right. \end{aligned}$$

(A.3)

The solution of (A.3) can be written explicitly,

$$\begin{aligned} \theta = -u^0(-1) \exp (-\bar{x}), \end{aligned}$$

(A.4)

and we expect $u^\varepsilon \approx u^0 + \theta$. The rigorous analysis like a convergence analysis can be found in the references introduced in this “Appendix”. The function $e^{-\frac{x+1}{\varepsilon }}$ represents the stiffness part of the boundary layer and it is defined as a corrector.

1.2 Example for the Interior Layer

For convection–diffusion equations which contain a turning point, we consider

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon \partial ^2_x u^\varepsilon - b(x) \partial _x u^\varepsilon = f, \quad x \in (-1,1),\\ u^\varepsilon (-1) = u^\varepsilon (1) = 0, \end{array}\right. } \end{aligned}$$

(A.5)

where

$$\begin{aligned} b< 0 \text { for } x<0, \quad b(0) = 0, \quad b> 0 \text { for } x>0, \end{aligned}$$

(A.6)

and $b'(x)> 0$. The interior layer arises at a turning point $x=0$ where the sign of b(x) changes and the characteristics collide.

For the formal limit problem, we set $\varepsilon =0$ in the above equation. Since $b(0)=0$, $\partial _x u^0$ may not be defined at $x=0$ and we thus consider the left and right parts ($u^0_l,u^0_r$, respectively) of $u^0$ from $x=0$,

$$\begin{aligned} -b(x) \partial _x u^0_l = f \quad \text { for } x < 0 \quad \text { and }\quad -b(x) \partial _x u^0_r = f \quad \text { for } x > 0. \end{aligned}$$

(A.7)

The inflow boundary conditions are then supplemented,

$$\begin{aligned} u^0_l(-1) = 0,\qquad u^0_r(1) = 0. \end{aligned}$$

(A.8)

The discrepancy at $x=0$ between $u^0_l$ and $u^0_r$ produce an interior layer. If $f(0) = 0$, the discrepancy is finite and thus the correctors introduced below can well capture the sharpness of interior layers. However, if $f(0) \ne 0$, the limit problem

$$\begin{aligned} -b(x) \partial _x u^0 = f \end{aligned}$$

(A.9)

has an inconsistency at $x=0$ since $b(0) = 0$. This implies that $\partial _x u^0$ blows up near $x=0$ where the interior layers cannot be captured by corrector functions. To remedy this difficulty, one can lift $u^\varepsilon$ so that the given data can be compatible (see (2.33)). For the singular perturbation analysis of the non-compatible case, see e.g. [28].

Using $\bar{x} = x/\sqrt{\varepsilon }$ and $b(x) = b'(0)x + \frac{1}{2}b^{''}(\xi )x^2 = b^{'}(0)\bar{x}\sqrt{\varepsilon } + \mathcal {O}(\varepsilon )$, some $\xi$ with $|\xi |\le |x|$, we substitute into (A.5) with $f=0$ and obtain the leading differential operators,

$$\begin{aligned} -\partial ^2_{\bar{x}} \theta - b^{'}(0) \bar{x} \partial _{\bar{x}} \theta = 0, \end{aligned}$$

(A.10)

supplemented with the boundary conditions,

$$\begin{aligned} \theta \rightarrow \text {constants}\ \text {as } \bar{x} \rightarrow \pm \infty . \end{aligned}$$

(A.11)

The constants are accurately approximated by the standard spectral methods. A particular solution of (A.10) can be written explicitly,

$$\begin{aligned} \theta = \frac{2}{\sqrt{\pi }} \int ^{\bar{x} \sqrt{\frac{b^{'}(0)}{2}}}_{0} e^{-\tau ^2} d \tau = erf \left( \bar{x} \sqrt{\frac{b^{'}(0)}{2}} \right) , \end{aligned}$$

(A.12)

which serves as a corrector.