Reference Materials
Published on January 13, 2022 by Joe Schoonover
For those of you interested in diving into the literature on spectral element methods, I've got a few references that I've been using to build the content for this and future livestreams I'd like to share.
A great book
The first is David Kopriva's book "Implementing Spectral Methods for Partial Differential Equations". This is the book I learned from in grad school where I coincidentally took a spectral element methods course with Dr. Kopriva! This book provides a healthy balance of theory and practice to help you get started with spectral element methods in one and two dimensions.
D.A. Kopriva (2009), "Implementing Spectral Methods for Partial Differential Equations", https://link.springer.com/book/10.1007/978-90-481-2261-5
A paper on stability
Working with variable coefficient advection problems is one step closer from constant advection towards more complex problems like Burger's equation and Navier-Stokes. Variable coefficient advection introduces aliasing errors that unfortunately break the guarantee of stability (relative to constant advection). This paper looks at a cost-effective strategy, using the split-form equation, rather than the conservative form to obtain an energy stable formulation.
D.A. Kopriva and G. Gassner (2014), "An Energy Stable Discontinuous Galerkin Spectral Element Discretization for Variable Coefficient Advection Problems" , http://dx.doi.org/10.1137/130928650
How these resources relate to this week's livestream
This week, I'll discuss the derivation of the discretization that we're working with to convince you that we have a discretization consistent with the advection equations. Additionally, I'll introduce spectral accuracy and argue that we can set up a scenario to check for spectral accuracy. All of this work, so far, is based off of the theory and algorithms presented in the book "Implementing Spectral Methods for Partial Differential Equations"
In the relatively simple test case, we'll be working with straight-sided elements and constant advection, a scenario in which we can actually prove stability. Unfortunately, this proof breaks down for more complex scenarios. However, Kopriva and Gassner (2014) discusses this stability problem and proposes an intriguing method for guaranteeing stability for more these complex scenarios with isoparametric elements and non-constant advection coefficients using the split-form equations and the summation-by-parts property of DG-Gauss derivative matrices .
A great book
The first is David Kopriva's book "Implementing Spectral Methods for Partial Differential Equations". This is the book I learned from in grad school where I coincidentally took a spectral element methods course with Dr. Kopriva! This book provides a healthy balance of theory and practice to help you get started with spectral element methods in one and two dimensions.
D.A. Kopriva (2009), "Implementing Spectral Methods for Partial Differential Equations", https://link.springer.com/book/10.1007/978-90-481-2261-5
A paper on stability
Working with variable coefficient advection problems is one step closer from constant advection towards more complex problems like Burger's equation and Navier-Stokes. Variable coefficient advection introduces aliasing errors that unfortunately break the guarantee of stability (relative to constant advection). This paper looks at a cost-effective strategy, using the split-form equation, rather than the conservative form to obtain an energy stable formulation.
D.A. Kopriva and G. Gassner (2014), "An Energy Stable Discontinuous Galerkin Spectral Element Discretization for Variable Coefficient Advection Problems" , http://dx.doi.org/10.1137/130928650
How these resources relate to this week's livestream
This week, I'll discuss the derivation of the discretization that we're working with to convince you that we have a discretization consistent with the advection equations. Additionally, I'll introduce spectral accuracy and argue that we can set up a scenario to check for spectral accuracy. All of this work, so far, is based off of the theory and algorithms presented in the book "Implementing Spectral Methods for Partial Differential Equations"
In the relatively simple test case, we'll be working with straight-sided elements and constant advection, a scenario in which we can actually prove stability. Unfortunately, this proof breaks down for more complex scenarios. However, Kopriva and Gassner (2014) discusses this stability problem and proposes an intriguing method for guaranteeing stability for more these complex scenarios with isoparametric elements and non-constant advection coefficients using the split-form equations and the summation-by-parts property of DG-Gauss derivative matrices .