3.2 The Diffusion Equation

In this section, the discretization of parabolic time-variant problems is described. In its simplest representation, the right-hand side of the diffusion equation is time-invariant. Solving a diffusion problem the diffusion equation becomes time-variant. It describes the out-diffusion of matter, driven by its own concentration gradient. The diffusion flux can be written as

where denotes the diffusion coefficient and is the concentration of the diffusing material.

Additionally, the conservation of material must be fulfilled

After insertion of (3.46) in (3.45), the diffusion equation can be reformulated as

(3.47) |

As described in the previous section, this equation can be discretized as

(3.48) |

The time derivative in this formula can be discretized by several methods. By the backward Euler method, the time derivative is approximated by [66]

(3.49) |

with the sampling interval. The discrete notation by backward Euler time discretization follows by

(3.50) |

By separating the unknowns to the left-hand side, this expression becomes

In matrix notation, (3.51) can be written as

with

(3.53) |

The necessary boundary conditions for this parabolic equation can be handled as shown in the previous section. Also the requirements for an M-matrix (see Section 2.3) are satisfied. The conditions for can be handled as in the previous section and consists of positive diagonal entries only. Additionally, an initial condition is required.

The initial concentration distribution at initial time is defined as

(3.60) |

The discrete system is satisfied by the discrete formulation

for all grid points | (3.61) |

The concentration distributions can be computed by an sequential evaluation of .

J. Cervenka: Three-Dimensional Mesh Generation for Device and Process Simulation