By Jayathi Y. Murthy
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Extra info for Numerical Methods in Heat, Mass, and Momentum Transfer (draft)
Though they yield discrete equations that look similar to other methods for simple cases, they are not guaranteed to do so in more complicated cases, for example on unstructured meshes. 1. There are different kinds of finite element methods. Let us look at a popular variant, the Galerkin finite element method. Let φ be an approximation to φ . 10 requires that the residual R become zero in a weighted sense. In order to generate a set of discrete equations we use a family of 30 ✂ ✓ ✓✰✯✱✯✱✯ weight functions Wi , i 1 2 N, where N is the number of grid points, rather than a single weight function.
51 for a few limiting cases. 1 The Explicit Scheme ✂ If we set f 0, we obtain the explicit scheme. This means that the flux and source terms are evaluated using values exclusively from the previous time step. 54) We notice the following about the discretization: 1. 53 contains values exclusively from the previous time t. Thus, given the condition at time t, it is possible for us to evaluate the right hand side completely, and find the value of φ P at time t ∆t. ✄ 47 2. We do not need to solve a set of linear algebraic equations to find φ P .
Thus, mixed boundaries 1. aP are like Dirichlet boundaries in that the boundary condition helps ensure that the Scarborough criterion is met. 2. The cell value φ P is bounded by its neighbor values φ E , φN and φS , and the external value, φ ∞ . 3. 34 once a solution has been obtained. 34. 1: ☎ ✕✝ ✄ ☎ We are given initial conditions φ x ✓ y✓ 0 ✝ . 39) As we saw in a previous chapter, time is a “marching” coordinate. By knowing the initial condition, and taking discrete time steps ∆t, we wish to obtain the solution for φ at a each discrete time instant.