A Nonoscillatory Numerical Scheme Based on A General Solution
for Unsteady Advection-Diffusion Equations
Katsuhiro Sakai
Saitama Institute of Technology,
Dept. of Electrical Engineering,
1690 Fusaiji, Okabe, Saitama,
369-02 Japan
E-mail:sakai@sit.ac.jp
Abstract
We consider the numerical solution of an unsteady advection-diffusion equation.
Conventional high-order numerical schemes for this equation tend
to show numerical oscillations. These oscillations are due to discretizing the
derivative terms in the equation.
In this paper, we apply a spectral technique to the unsteady
advection-diffusion equation with a spatially uniform velocity
field. The solution f(t,x) is uniquely expanded into the Fourie
series using orthonormal and complete functions in a finite
computational region.
Substituting the f(t,x) expressed in the Furie series into the
advection-diffusion equation yields the ordinary differential
equation for the transformed function C(t,k) of f(t,x).
In this course, both of the temporal and spatial derivative terms
are treated with analytically, resulting in being free from the
deficiency due to discretizing the derivative terms. Given C(t',k)
at the time t' as an initial condition, we solve the ordinary
differential equation and obtain C(t,k) with C(t',k), which is
given by the Fourie transformation of f(t',x') at t'.
Substituting this C(t,k) into C(t,k) in f(t,x) expressed in
the Fourie series, and performing an algebraic manipulation using a
formula about the sum in an infinite series with respect to k,
we obtain a new expression for f(t,x) in terms of f(t',x') and a
correlation function between the two solutions at (t',x') and
at (t,x). In numerical calculations, a discrete approximation
with trapezoidal rule or the Simpson's rule for the
integration with respect to x' in this expression is performed.
When the diffusivity goes to 0, this expression reduces to a
usual advection properties of hyperbolic equation owing to the
nature of the delta function.
This new expression is a formula to calculate f(t,x)
explicitly by using f(t',x') at the older time t'. Namely,
given the distribution of a quantity in the real space at one
step older time, we solve the quantity at new time step still
in the real space, in which the spectral technique is applied
only to it's deviation from the quantity at older time step.
In this point of view, the present method is not always same
as the usual spectral method. The sum with respect to the
Fourie mode k in the present formula tends to converge rapidly
thanks to the exponential function.
Moreover, the coefficients associated with f(t',x')
in the right-hand side of the present formula are positive at
any time and anywhwere for any large velocity and for any small
diffusivity. Hence the present formula fulfills the Patanker's
positive coefficients condition. Hence this formula guarantees
solutions free from numerical oscillations for unsteady advection-diffusion
equations with any large velocity and any small diffusivity.
Thus, a nonoscillatory numerical scheme was presented.
The present scheme can be straightforwardly
extended to multidimensional problems. Numerical experiments
showed good solutions without numerical oscillations.