Open Access. Powered by Scholars. Published by Universities.®
- Publication Type
Articles 1 - 2 of 2
Full-Text Articles in Applied Mathematics
Sparse Spectral-Tau Method For The Two-Dimensional Helmholtz Problem Posed On A Rectangular Domain, Gabriella M. Dalton
Sparse Spectral-Tau Method For The Two-Dimensional Helmholtz Problem Posed On A Rectangular Domain, Gabriella M. Dalton
Mathematics & Statistics ETDs
Within recent decades, spectral methods have become an important technique in numerical computing for solving partial differential equations. This is due to their superior accuracy when compared to finite difference and finite element methods. For such spectral approximations, the convergence rate is solely dependent on the smoothness of the solution yielding the potential to achieve spectral accuracy. We present an iterative approach for solving the two-dimensional Helmholtz problem posed on a rectangular domain subject to Dirichlet boundary conditions that is well-conditioned, low in memory, and of sub-quadratic complexity. The proposed approach spectrally approximates the partial differential equation by means of …
On The Consistency Of Alternative Finite Difference Schemes For The Heat Equation, Tran April
On The Consistency Of Alternative Finite Difference Schemes For The Heat Equation, Tran April
Rose-Hulman Undergraduate Mathematics Journal
While the well-researched Finite Difference Method (FDM) discretizes every independent variable into algebraic equations, Method of Lines discretizes all but one dimension, leaving an Ordinary Differential Equation (ODE) in the remaining dimension. That way, ODE's numerical methods can be applied to solve Partial Differential Equations (PDEs). In this project, Linear Multistep Methods and Method of Lines are used to numerically solve the heat equation. Specifically, the explicit Adams-Bashforth method and the implicit Backward Differentiation Formulas are implemented as Alternative Finite Difference Schemes. We also examine the consistency of these schemes.