Improving the Finite Difference Approximation in the Jacobian-Free Newton–Krylov Method

Sammanfattning: The Jacobian-free Newton–Krylov (JFNK) method is designed to solve a linear system of equations that appears in Newton’s method. It uses the generalized minimal residual (GMRES) method to solve the linear system and a simple function to approximate the matrix-vector multiplications required in GMRES. An advantage of GMRES is the ability to check the residual of a potential solution without doing any extra computations. A previous bachelor’s thesis discovered that the residual seen by the algorithm and the actual residual differ across many test cases. This puts the validity of the solution into question and makes it difficult to implement any sort of error-checking in the algorithm. The purpose of this thesis is to investigate the discrepancy between these residuals. Tests were run on the various problems and although they were unable to determine a concrete explanation for the behavior of the residuals, they provided valuable insight into the potential causes to investigate in the future.