By Hans Petter Langtangen
This textual content presents an easy, preliminary advent to the whole medical computing pipeline: types, discretization, algorithms, programming, verification, and visualization. The pedagogical technique is to exploit one case research – a standard differential equation describing exponential decay tactics – to demonstrate basic options in arithmetic and laptop technological know-how. The publication is simple to learn and purely calls for a command of one-variable calculus and a few very simple wisdom approximately laptop programming. opposite to related texts on numerical tools and programming, this article has a far better specialise in implementation and teaches checking out and software program engineering particularly.
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Extra resources for Finite Difference Computing with Exponential Decay Models
This function evaluates -a*t, which is a scalar times an array, meaning that the scalar is multiplied with each array element. The result is an array, let us call it tmp1. Then exp(tmp1) means applying the exponential function to each element in tmp1, giving an array, say tmp2. py 26 1 Algorithms and Implementations Finally, I*tmp2 is computed (scalar times array) and u_e refers to this array returned from u_exact. The expression u_e - u is the difference between two arrays, resulting in a new array referred to by e.
Representative of exact sol. Note that the mesh functions u and u_e are represented by arrays and associated with the points in the array t. Array arithmetics The last statements u_e = u_exact(t, I, a) e = u_e - u demonstrate some standard examples of array arithmetics: t is an array of mesh points that we pass to u_exact. This function evaluates -a*t, which is a scalar times an array, meaning that the scalar is multiplied with each array element. The result is an array, let us call it tmp1. Then exp(tmp1) means applying the exponential function to each element in tmp1, giving an array, say tmp2.
The most obvious idea for verification in our case is to compare the numerical solution with the exact solution, when that exists. This is, however, not a particularly good method. The reason is that there will always be a discrepancy between these two solutions, due to numerical approximations, and we cannot precisely quantify the approximation errors. The open question is therefore whether we have the mathematically correct discrepancy or if we have another, maybe small, discrepancy due to both an approximation error and an error in the implementation.
Finite Difference Computing with Exponential Decay Models by Hans Petter Langtangen