Application of sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems. (English) Zbl 1301.65108
This paper proposes a numerical method to solve singularly perturbed parabolic initial boundary value problems (IBVPs) of the form:
\[
\begin{gathered}\dfrac{\partial u}{\partial t} - \varepsilon \dfrac{\partial^2 u}{\partial x^2} + a(x,t) \dfrac{\partial u}{\partial x} + b(x,t) u = f(x,t), \quad (x,t) \in (0,1) \times (0,T),\\ u(x,0) = u_0(x),\quad x \in (0,1), \\ u(0,t) = g_0(t), \quad u(1,t) = g_1(t), \quad t \in (0,T), \end{gathered}
\]
where \(0 < \varepsilon \ll 1\).
It is well known that the solution of the above IBVP exhibits regular boundary layers, and one has to use layer-adapted meshes to solve these kinds of problems.
Here, the authors use the method of lines to semidiscretize the parabolic PDE in time by using the backward Euler scheme, and obtain a system of boundary value problems at each time level. The resultant BVPs are solved by applying the sinc-Galerkin method. The authors obtain some mathematical results showing the stability and convergence of the present method.
It is clear from the stability result that the present method is not \(\varepsilon\)-uniform stable, because the stability inequality contains the diffusion parameter \(\varepsilon\) in the denominator. In general, one is interested to study the above IBVP, for smaller values of \(\varepsilon\). In this case, the other terms in the stability inequality should be adjusted, which is not practical.
Therefore, the resulting error estimates are not \(\varepsilon\)-uniformly convergent. This can be easily seen from the error tables provided in the numerical examples. The earlier methods, which are used for comparison are all \(\varepsilon\)-uniform convergent (this can be easily seen, as \(\varepsilon \to 0\) the maximum pointwise error remains the same), whereas in the present method the maximum pointwise-error increases as \(\varepsilon\) becomes smaller.
It is evident that layer-adapted meshes are necessary to obtain \(\varepsilon\)-uniform convergence solutions for classical finite difference/element methods. This is applicable to the present method as well.
It is well known that the solution of the above IBVP exhibits regular boundary layers, and one has to use layer-adapted meshes to solve these kinds of problems.
Here, the authors use the method of lines to semidiscretize the parabolic PDE in time by using the backward Euler scheme, and obtain a system of boundary value problems at each time level. The resultant BVPs are solved by applying the sinc-Galerkin method. The authors obtain some mathematical results showing the stability and convergence of the present method.
It is clear from the stability result that the present method is not \(\varepsilon\)-uniform stable, because the stability inequality contains the diffusion parameter \(\varepsilon\) in the denominator. In general, one is interested to study the above IBVP, for smaller values of \(\varepsilon\). In this case, the other terms in the stability inequality should be adjusted, which is not practical.
Therefore, the resulting error estimates are not \(\varepsilon\)-uniformly convergent. This can be easily seen from the error tables provided in the numerical examples. The earlier methods, which are used for comparison are all \(\varepsilon\)-uniform convergent (this can be easily seen, as \(\varepsilon \to 0\) the maximum pointwise error remains the same), whereas in the present method the maximum pointwise-error increases as \(\varepsilon\) becomes smaller.
It is evident that layer-adapted meshes are necessary to obtain \(\varepsilon\)-uniform convergence solutions for classical finite difference/element methods. This is applicable to the present method as well.
Reviewer: Srinivasan Natesan (Assam)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35B25 | Singular perturbations in context of PDEs |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
Keywords:
singular perturbation; semidiscretization; sinc-Galerkin method; stability; convergence; parabolic initial boundary value problem; boundary layer; method of lines; backward Euler schemeReferences:
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