TY - JOUR
T1 - A New High Order Two Level Implicit Discretization for the Solution of 3D Nonlinear Parabolic Equations
AU - R. K. Mohanty & S. Singh
JO - International Journal of Numerical Analysis and Modeling
VL - 1
SP - 40
EP - 54
PY - 2008
DA - 2008/05
SN - 5
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/796.html
KW - non-linear parabolic equation
KW - implicit scheme
KW - high order method
KW - normal derivatives
KW - singular problem
KW - operator splitting
KW - Burgers' equation
AB - We present a new two-level implicit difference method of
O(k(2) + kh(2) + h(4)) for approximating the three space dimensional
non-linear parabolic differential equation u(xx) +u(yy) +u(zz) = f(x, y,
z, t, u, u(x), u(y), u(z), u(t)), 0 < x, y, z < 1, t > 0 subject to
appropriate initial and Dirichlet boundary conditions, where h > 0 and k
> 0 are mesh sizes in space and time directions, respectively. In
addition, we also propose some new two-level explicit stable methods of
O(kh(2) + h(4)) for the estimates of (partial derivative u/partial
derivative n). When grid lines are parallel to x-, y- and z- coordinate
axes, then (partial derivative u/partial derivative n) at an internal
grid point becomes (partial derivative u/partial derivative x), (partial
derivative u/partial derivative y) and (partial derivative u/partial
derivative z), respectively. In all cases, we require only 19-spatial
grid points and a single computational cell. The proposed methods are
directly applicable to singular problems and we do not require any
special technique to handle singular problems. We also discuss operator
splitting method for solving linear parabolic equation. This method
permits multiple use of the one-dimensional tri-diagonal solver. It is
shown that the operator splitting method is unconditionally stable.
Numerical tests are conducted which demonstrate the accuracy and
effectiveness of the methods developed.