The characteristic variational multiscale method for
convection-dominated convection–diffusion–reaction problems
Lingzhi Qian
a,b
, Huiping Cai
b
, Rui Guo
b
, Xinlong Feng
c,
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a
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR China
b
Department of Mathematics, College of Sciences, Shihezi University, Shihezi 832003, PR China
c
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, PR China
article info
Article history:
Received 8 February 2013
Received in revised form 4 December 2013
Accepted 6 January 2014
Available online 4 February 2014
Keywords:
Convection-dominated diffusion equation
Characteristics
Variational multiscale method
Stability analysis
Error estimate
abstract
In this paper, the characteristic variational multiscale (C-VMS) method is proposed for solving
two-dimensional (2D) convection-dominated convection–diffusion–reaction problems. The scheme is
combined the method of characteristics with the variational multiscale (VMS) method to create the C-
VMS procedures. The stability analysis and error estimate of the C-VMS method are obtained. The scheme
not only realizes the purpose of lowering the time-truncation error, using larger time step for solving the
convection-dominated convection–diffusion–reaction problems, but also keeps the favorable stability
and high precision. Finally, numerical experiments in 2D and 3D cases are presented to illustrate the
availability and efficiency of the scheme.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
It is well known that the convection-dominated diffusion prob-
lem has strongly hyperbolic nature, the solution often develops
sharp fronts that are nearly shocks. So the finite difference method
(FDM) or finite element method (FEM) applied to the problem do
not work well when it is convection-dominated. Therefore to con-
struct an effective numerical method for solving such a problem is
not easy. In 1982, Douglas and Russell [2] considered combining
the method of characteristics with finite element or finite differ-
ence techniques to overcome oscillation and faults likely to occur
in the traditional FDM or FEM. There are many related approxima-
tion techniques for the convection-dominated diffusion equations,
for examples, Tabata and his coworkers [1,24,25] have been stud-
ied the upwind schemes based on triangulation for the convection–
diffusion problems. Yuan presented a characteristic finite element
alternating direction method with moving meshes [30] and an
upwind finite difference fractional step method [29], respectively.
Gao and Yuan [5] used the upwind finite volume element method
based on straight triangular prism partition for the nonlinear
convection–diffusion problems. Qian et al. [17] presented the char-
acteristic finite difference streamline diffusion method for convec-
tion-dominated convection–diffusion–reaction problems. Si et al.
[20] proposed the semi-discrete streamline diffusion FEM for the
nonstationary convection–diffusion problems. In problems with
significant convection, these approaches of characteristics have
much smaller time-truncation errors compared to FDM or FEM.
Moreover, these schemes of characteristics will permit the use of
larger time steps, with corresponding improvements in efficiency,
at no cost in accuracy [5,6,8,9,18,22,30].
The variational multiscale (VMS) method is introduced by
Hughes [11], which is motivated by the inherent multiscale struc-
ture of the solution. The VMS method can be viewed as a procedure
for deriving models and numerical methods capable of overcoming
spurious oscillations in solutions cased by the multiscale method.
The VMS method decompose the solution into large scale and
small subgrid scale and write problem as a coupled system of
two sub-problems for the two types of scales. The difficulty of
VMS method lies in the solving small subgrid scale solution whose
effect on large scale solution always be nonlocal [12]. To overcome
such difficulty lots of methods have been developed, e.g. the resid-
ual free bubble method [4,16] and the subgrid stabilization method
[7,13–15]. Other noteworthy contributions for the convection–dif-
fusion problems are due to Wu et al. [28], in order to eliminate
overshoots and undershoots produced by the convection term in
the meshless local Petrov–Galerkin method, they used the stream-
line upwind Petrov–Galerkin to solve the convection-dominated
problems. Sheu and Lin [19] applied the fourth-order-accurate
temporal/spatial scheme to the convection–diffusion equations.
Zhai et al. [31] proposed some new high-order compact difference
schemes for the stationary semilinear convection–diffusion
0017-9310/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.01.020
⇑
Corresponding author. Tel./fax: +86 991 8582482.
E-mail addresses: fxlmath@gmail.com, fxlmath@xju.edu.cn (X. Feng).
International Journal of Heat and Mass Transfer 72 (2014) 461–469
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
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