Interested in learning how to solve partial differential equations with numerical methods and how to turn them into python codes? This course provides you with a basic introduction how to apply methods like the finite-difference method, the pseudospectral method, the linear and spectral element method to the 1D (or 2D) scalar wave equation. The mathematical derivation of the computational algorithm is accompanied by python codes embedded in Jupyter notebooks. In a unique setup you can see how the mathematical equations are transformed to a computer code and the results visualized. The emphasis is on illustrating the fundamental mathematical ingredients of the various numerical methods (e.g., Taylor series, Fourier series, differentiation, function interpolation, numerical integration) and how they compare. You will be provided with strategies how to ensure your solutions are correct, for example benchmarking with analytical solutions or convergence tests. The mathematical aspects are complemented by a basic introduction to wave physics, discretization, meshes, parallel programming, computing models.
The course targets anyone who aims at developing or using numerical methods applied to partial differential equations and is seeking a practical introduction at a basic level. The methodologies discussed are widely used in natural sciences, engineering, as well as economics and other fields.
What you will learn
Week 01 – Discrete World, Wave Physics, Computers
The use of numerical methods to solve partial differential equations is motivated giving examples form Earth sciences. Concepts of discretization in space and time are introduced and the necessity to sample fields with sufficient accuracy is motivated (i.e. number of grid points per wavelength). Computational meshes are discussed and their power and restrictions to model complex geometries illustrated. The basics of parallel computers and parallel programming are discussed and their impact on realistic simulations. The specific partial differential equation used in this course to illustrate various numerical methods is presented: the acoustic wave equation. Some physical aspects of this equation are illustrated that are relevant to understand its solutions. Finally Jupyter notebooks are introduced that are used with Python programs to illustrate the implementation of the numerical methods.
Week 02 The Finite-Difference Method – Taylor Operators
In Week 2 we introduce the basic definitions of the finite-difference method. We learn how to use Taylor series to estimate the error of the finite-difference approximations to derivatives and how to increase the accuracy of the approximations using longer operators. We also learn how to implement numerical derivatives using Python.
Week 03 The Finite-Difference Method – 1D Wave Equation – von Neumann Analysis
We develop the finite-difference algorithm to the acoustic wave equation in 1D, discuss boundary conditions and how to initialize a simulation example. We look at solutions using the Python implementation and observe numerical artifacts. We analytically derive one of the most important results of numerical analysis – the CFL criterion which leads to a conditionally stable algorithm for explicit finite-difference schemes.
Week 04 The Finite-Difference Method in 2D – Numerical Anisotropy, Heterogeneous Media
We develop the solution to the 2D acoustic wave equation, compare with analytical solutions and demonstrate the phenomenon of numerical (non-physical) anisotropy. We extend the von Neumann Analysis to 2D and derive numerical anisotropy analytically. We learn how to initialize a realistic physical problem and illustrate that 2D solution are already quite powerful to understand complex wave phenomena. We introduced the 1D elastic wave equation and show the concept of staggered-grid schemes with the coupled first-order velocity-stress formulation.