An integral equation approach to the unsteady convection-diffusion equations

An integral equation approach is proposed to solve the unsteady convection-diffusion equations. In this approach, the second order Adams-Moulton method is firstly utilized for the time discretization. Then by using the Green's function of the Laplace equation in the series form, the convection-diffusion equation is transformed into an integral equation that is further converted into an algebraic equation system. The accuracy, convergence and the stability of this integral equation approach are examined by four examples. In comparison with the characteristic variational multiscale method and the finite volume element method, the integral equation approach shows a higher accuracy. Compared with the finite volume element method the integral equation approach has a better convergence. In solving the convection dominated convection-diffusion problems the integral equation approach demonstrates a good stability.

[1]  W. Bechteler,et al.  Practical Aspects for the Application of the Diffusion-Convection Theory for Sediment Transport in Turbulent Flows , 1988 .

[2]  许明田.  Effect of soft-iron impellers on the von Karman-sodium dynamo , 2014 .

[3]  Tosawat Seetawan,et al.  Analysis of Thermoelectric Generator by Finite Element Method , 2012 .

[4]  Klaus-Jürgen Bathe,et al.  A finite element method enriched for wave propagation problems , 2012 .

[5]  D. Rose,et al.  Some errors estimates for the box method , 1987 .

[6]  Jose Alvarez-Ramirez,et al.  Nonstandard finite difference schemes based on Green's function formulations for reaction–diffusion–convection systems , 2013 .

[7]  Kyuro Sasaki,et al.  A grid-free particle tracking simulation for tracer dispersion in porous reservoir model , 2015 .

[8]  Wei Liu,et al.  The MMOC and MMOCAA schemes for the finite volume element method of convection-diffusion problems , 2011, J. Comput. Appl. Math..

[9]  Tong Zhang,et al.  The semidiscrete finite volume element method for nonlinear convection-diffusion problem , 2011, Appl. Math. Comput..

[10]  Patricia L. Wiberg,et al.  A two-dimensional, time-dependent model of suspended sediment transport and bed reworking for continental shelves , 2001 .

[11]  Markus Bause,et al.  Analysis of stabilized higher-order finite element approximation of nonstationary and nonlinear convection–diffusion–reaction equations , 2012 .

[12]  F. Stefani,et al.  The integral equation method for a steady kinematic dynamo problem , 2004 .

[13]  Zhiqiang Cai,et al.  On the finite volume element method , 1990 .

[14]  Garrett Birkhoff,et al.  Difference methods for solving convection-diffusion equations , 1990 .

[15]  Peter Szmolyan,et al.  A system of convection—diffusion equations with small diffusion coefficient arising in semiconductor physics , 1989 .

[16]  Yintzer Shih,et al.  An exponential-fitting finite element method for convection-diffusion problems , 2011, Appl. Math. Comput..

[17]  Xinlong Feng,et al.  The characteristic variational multiscale method for convection-dominated convection–diffusion–reaction problems , 2014 .

[18]  Jürgen Geiser,et al.  Error estimates for finite volume element methods for convection-diffusion-reaction equations , 2007 .

[19]  Mingtian Xu,et al.  Integral equation approach to time-dependent kinematic dynamos in finite domains. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Theodosios Korakianitis,et al.  Finite integration method for partial differential equations , 2013 .

[21]  M. Farhloul,et al.  A mixed-hybrid finite element method for convection-diffusion problems , 2005, Appl. Math. Comput..

[22]  Adem Kaya,et al.  A finite difference scheme for multidimensional convection-diffusion-reaction equations , 2014 .

[23]  Matthias Ehrhardt,et al.  A nonstandard finite difference scheme for convection-diffusion equations having constant coefficients , 2013, Appl. Math. Comput..

[24]  S. Adeeb,et al.  The finite element method for micro-scale modeling of ultrasound propagation in cancellous bone. , 2014, Ultrasonics.

[25]  S. Nicaise,et al.  A non-conforming finite volume element method of weighted upstream type for the two-dimensional stationary Navier--Stokes system , 2008 .

[26]  Tao Wei,et al.  An integral equation approach to two-dimensional incompressible resistive magnetohydrodynamics , 2014 .

[27]  Pavel A. Strizhak,et al.  Transient heat and mass transfer at the ignition of vapor and gas mixture by a moving hot particle , 2010 .

[28]  K. Bathe.  Finite Element Procedures , 1995 .

[29]  Yirang Yuan,et al.  A symmetric characteristic FVE method with second order accuracy for nonlinear convection diffusion problems , 2007 .

[30]  A. H. Schatz,et al.  Crosswind Smear and Pointwise Errors in Streamline Diffusion Finite Element Methods , 1987 .

[31]  Sanjeev Kumar Raghuwanshi,et al.  Dispersion characteristics of complex refractive-index planar slab optical waveguide by using finite element method , 2014 .

[32]  Huanrong Li.  Analysis and application of finite volume element methods to a class of partial differential equations , 2009 .

[33]  A. Wit,et al.  Influence of buoyancy-driven convection on the dynamics of A+ B → C reaction fronts in horizontal solution layers , 2010 .

[34]  Ya-Ling He,et al.  Some recent advances in finite volume approach and their applications in the study of heat transfer enhancement , 2005 .

[35]  F. Stefani,et al.  Ambivalent effects of added layers on steady kinematic dynamos in cylindrical geometry: application to the VKS experiment , 2006 .

[36]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[37]  Mingtian Xu,et al.  The integral equation approach to kinematic dynamo theory and its application to dynamo experiments in cylindrical geometry , 2006, J. Comput. Phys..

[38]  Zhiqiang Cai,et al.  The finite volume element method for diffusion equations on general triangulations , 1991 .

[39]  A van Niekerk,et al.  An explicit finite element method for convection-diffusion equations using rational basis functions , 1990 .

[40]  Xiaohua Yang,et al.  Chaos gray-coded genetic algorithm and its application for pollution source identifications in convection–diffusion equation , 2008 .

[41]  Maria Antonietta Pirozzi,et al.  High order finite difference numerical methods for time-dependent convection-dominated problems , 2005 .

[42]  Montri Maleewong,et al.  Finite Element Method for Dissolved Oxygen and Biochemical Oxygen Demand in an Open Channel , 2012 .