By Hans Petter Langtangen
This textual content presents an easy, preliminary creation to the whole medical computing pipeline: types, discretization, algorithms, programming, verification, and visualization. The pedagogical process is to exploit one case research – a regular differential equation describing exponential decay procedures – to demonstrate primary thoughts in arithmetic and machine technological know-how. The publication is straightforward to learn and in simple terms calls for a command of one-variable calculus and a few very uncomplicated wisdom approximately computing device programming. opposite to comparable texts on numerical tools and programming, this article has a far more advantageous concentrate on implementation and teaches trying out and software program engineering particularly.
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Extra resources for Finite Difference Computing with Exponential Decay Models
In fact, une is often called the representative of ue on the mesh. Then, e n D une un is clearly the difference of two mesh functions. The error mesh function e n can be computed by u, t = solver(I, a, T, dt, theta) u_e = u_exact(t, I, a) e = u_e - u # Numerical sol. # 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.
10. The L2 norm of the error can be computed by treating e n as a function of t in sympy and performing symbolic integration. 3 Accuracy 51 notify ourselves that the functions are from sympy. sin is for symbolic expressions, while sin from numpy or math is for numerical computation.
Also the sqrt function is from numpy and computes the square root of each element in the array argument. size) = zeros(m) 0 i in range(m): u_e[i] = u_exact(t, a, I) t = t + dt e = zeros(m) for i in range(m): e[i] = u_e[i] - u[i] s = 0 # summation variable for i in range(m): s = s + e[i]**2 error = sqrt(dt*s) Such element-wise computing, often called scalar computing, takes more code, is less readable, and runs much slower than what we can achieve with array computing. 11 Experiments with Computing and Plotting Let us write down a new function that wraps up the computation and all the plotting statements used for comparing the exact and numerical solutions.