1 Two-dimensional heat equation with FD

Excerpt from GEOL557 Numerical Modeling of Earth Systems by Becker and Kaus (2016). i+1,j i,j-1 i,j i,j+1. H. z i-1,j Dz Dx L. x Figure 1: Finite difference discretization of the 2D heat problem. 1 Two-dimensional heat equation with FD. We now revisit the transient heat equation , this time with sources/sinks, as an example for Two-dimensional FD problem. In 2D ({ x, z} space), we can write . T T T. c p = kx + kz +Q (1). t x x z z where, is density, c p heat capacity, k x,z the thermal conductivities in x and z direction, and Q radiogenic heat production. If the thermal conductivity is isostropic (k x = k z ) and constant, we can rewrite 2 T 2 T.. T Q. = + 2 + . (2). t x2 z c p Explicit method The simplest way to discretize eq.

We now revisit the transient heat equation, this time with sources/sinks, as an example for two-dimensional FD problem. In 2D (fx,zgspace), we can write rcp ¶T ¶t = ¶ ¶x kx ¶T ¶x + ¶ ¶z kz ¶T ¶z +Q (1) where, r is density, cp heat capacity, kx,z the thermal conductivities in x and z direction, and Q radiogenic heat production.

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  Heat, Equations, Heat equation

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