A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equ
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A uniformly convergent quadratic B‑spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters Meenakshi Shivhare1 · Pramod Chakravarthy Podila1 · Devendra Kumar2 Received: 7 June 2020 / Accepted: 14 October 2020 © Springer Nature Switzerland AG 2020
Abstract This paper aims to construct a parameters-uniform numerical scheme to solve the singularly perturbed parabolic partial differential equations whose solution exhibits parabolic (or exponential) boundary layers at both the lateral surfaces of the rectangular domain. The method comprises an implicit Euler scheme on a uniform mesh in the temporal direction and the quadratic B-spline collocation scheme on an exponentially graded mesh in the spatial direction. The exponentially graded mesh is generated by choosing an appropriate mesh generating function which adapts the mesh points in the boundary layers appear in the spatial direction. To establish the error estimates the solution is decomposed into its regular and singular components and the error estimates for these components are obtained separately. We prove the parameters-uniform convergence of the proposed numerical scheme and the method is shown to be of O(Nx−2 + Δt) where Nx denotes the number of mesh points in the space direction and Δt is the mesh step size in the temporal direction. To support the obtained theoretical estimates, two test examples are considered numerically. Keywords Singular perturbation · Parabolic partial differential equations · Twoparameter · Collocation method · Quadratic B-splines · Exponentially graded mesh · Error analysis Mathematics Subject Classification 35K20 · 65M12 · 65M15 · 65M22 · 65M70 * Pramod Chakravarthy Podila [email protected] Meenakshi Shivhare [email protected] Devendra Kumar [email protected]‑pilani.ac.in 1
Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India
2
Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan 333031, India
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Vol.:(0123456789)
Journal of Mathematical Chemistry
1 Introduction Singularly perturbed problems (SPPs) arise in many phenomena of engineering, biology, physics described by boundary layer problems with the various type of ordinary and partial differential equations. SPPs with two small parameters model in the chemical reactor theory [1], flow through unsaturated porous media [2], lubrication theory [3–5], the hydrological situation of 1D vertical groundwater, transport phenomena [6], and so forth. These problems behave a multi-scale character where the solution has a sudden change in some narrow region of the domain called the boundary layer region and the solution is uniform otherwise. Due to which standard numerical methods on a uniform mesh do not work properly and lead to an oscillatory solution and thus fails to give the required results. For obtaining the uniform solution very fine mesh is required in the boundary layer region with the number of mesh points
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