Discretization Error Guided Optimal Mesh Adaptation in One-dimensional Steady Problems

Abstract

© 2023, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.When solving differential equations, one must often use spatial discretization. However, this process introduces errors that are mesh dependent. Thus, improving solution quality while saving computational resources requires adequate spatial resolution. One way of doing so is to treat this issue as an optimization problem that targets the reduction of discretization error. The current work presents an approach to mesh optimization using r-adaptation and the adjoint method for one-dimensional steady equations. The two equations selected to display this methodology are the heat equation with a forcing term and the viscous burgers equation. The discretization method is a second-order finite differences scheme. The results present a substantial reduction in discretization error when the optimized meshes are employed.