Volume 5, Issue 1
A New High Order Two Level Implicit Discretization for the Solution of 3D Nonlinear Parabolic Equations

R. K. Mohanty & S. Singh

Int. J. Numer. Anal. Mod., 5 (2008), pp. 40-54

Published online: 2008-05

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  • Abstract

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.

  • Keywords

non-linear parabolic equation implicit scheme high order method normal derivatives singular problem operator splitting Burgers' equation

  • AMS Subject Headings

35N 65N

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-5-40, author = {R. K. Mohanty and S. Singh}, title = {A New High Order Two Level Implicit Discretization for the Solution of 3D Nonlinear Parabolic Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2008}, volume = {5}, number = {1}, pages = {40--54}, abstract = {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. }, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/796.html} }
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.
R. K. Mohanty & S. Singh. (1970). A New High Order Two Level Implicit Discretization for the Solution of 3D Nonlinear Parabolic Equations. International Journal of Numerical Analysis and Modeling. 5 (1). 40-54. doi:
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