An analytic adjoint Trefftz method for solving the singular parabolic convection–diffusion equation and initial pollution profile problem

Abstract In this paper we develop a novel method for the solution of a singularly perturbed convection–diffusion equation by inserting the adjoint Trefftz functions as test functions into the derived domain/boundary integral equation. The trial solution is expressed in terms of the Trefftz series (i.e., the spatial part of the Trefftz test functions), endowing with time-dependent expansion coefficients. With the aid of the weighting orthogonality of the Trefftz series we can find these coefficients in closed-form, which are integrals of initial condition, boundary conditions and source function. Hence, we have an analytical method to find the singular solution, very close to the exact one if more series terms are used. On the other hand, when the recovery of initial pollution profile by using the measured data at a final time is concerned, the number of the Trefftz series must be suitably chosen to overcome the ill-posed behavior of the initial pollution profile problem. The present results are better than that obtained by the Lie-group shooting method.

[1]  Piet Hemker,et al.  ε-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems , 2000 .

[2]  Amvrossios C. Bagtzoglou,et al.  Pollution source identification in heterogeneous porous media , 2001 .

[3]  Carmelo Clavero,et al.  A high order uniformly convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems , 2007, Numerische Mathematik.

[4]  T. Linß.  Layer-adapted meshes and FEM for time-dependent singularly perturbed reaction-diffusion problems , 2007, Int. J. Comput. Sci. Math..

[5]  Srinivasan Natesan,et al.  Richardson extrapolation technique for singularly perturbed parabolic convection–diffusion problems , 2010, Computing.

[6]  Chih-Wen Chang,et al.  The backward group preserving scheme for 1D backward in time advection-dispersion equation , 2010 .

[7]  Nicholas Zabaras,et al.  A markov random field model of contamination source identification in porous media flow , 2006 .

[8]  Chih-Wen Chang,et al.  A Fictitious Time Integration Method for Backward Advection-Dispersion Equation , 2009 .

[9]  A. Bagtzoglou,et al.  State of the Art Report on Mathematical Methods for Groundwater Pollution Source Identification , 2001 .

[10]  On Finite Difference Fitted Schemes for Singularly Perturbed Boundary Value Problems with a Parabolic Boundary Layer , 1997 .

[11]  Carmelo Clavero,et al.  On the uniform convergence of a finite difference scheme for time dependent singularly perturbed reaction-diffusion problems , 2010, Appl. Math. Comput..

[12]  Carmelo Clavero,et al.  High order methods for elliptic and time dependent reaction-diffusion singularly perturbed problems , 2005, Appl. Math. Comput..

[13]  Finite difference domain decomposition algorithms for a parabolic problem with boundary layers , 1998 .

[14]  Chih-Wen Chang,et al.  A Quasi-Boundary Semi-Analytical Method for Backward in Time Advection-Dispersion Equation , 2009 .

[15]  Brian J. Wagner,et al.  Simultaneous parameter estimation and contaminant source characterization for coupled groundwater flow and contaminant transport modelling , 1992 .

[16]  Vikas Gupta,et al.  A brief survey on numerical methods for solving singularly perturbed problems , 2010, Appl. Math. Comput..

[17]  M. K. Kadalbajoo,et al.  A survey of numerical techniques for solving singularly perturbed ordinary differential equations , 2002, Appl. Math. Comput..

[18]  Chein-Shan Liu.  An LGSM to identify nonhomogeneous heat conductivity functions by an extra measurement of temperature , 2008 .

[19]  S. Gorelick,et al.  Identifying sources of groundwater pollution: An optimization approach , 1983 .

[20]  N. V. Kopteva.  On the uniform in small parameter convergence of a weighted scheme for the one-dimensional time-dependent convection-diffusion equation , 1997 .

[21]  J. C. Jorge,et al.  An alternating direction scheme on a nonuniform mesh for reaction-diffusion parabolic problems , 2000 .

[22]  Chein-Shan Liu,et al.  The Lie-Group Shooting Method for Nonlinear Two-Point Boundary Value Problems Exhibiting Multiple Solutions , 2006 .

[23]  Torsten Linß,et al.  Parameter uniform approximations for time‐dependent reaction‐diffusion problems , 2007 .

[24]  M. P. Rajan,et al.  An iterative technique for solving singularly perturbed parabolic PDE , 2016 .

[25]  Martin Stynes,et al.  Numerical methods for time-dependent convection-diffusion equations , 1988 .

[26]  Amvrossios C. Bagtzoglou,et al.  Marching‐jury backward beam equation and quasi‐reversibility methods for hydrologic inversion: Application to contaminant plume spatial distribution recovery , 2003 .

[27]  Chih-Wen Chang,et al.  A new shooting method for quasi-boundary regularization of backward heat conduction problems , 2007 .