# Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods

## TL;DR

Dedalus is an open-source spectral methods framework that efficiently solves various partial differential equations through a flexible, object-oriented design, enabling complex scientific simulations with high performance and customization.

## Contribution

This paper introduces Dedalus, a novel flexible spectral methods framework that automates PDE solving and supports diverse scientific problems with a modular, object-oriented implementation.

## Key findings

- Demonstrated high parallel scalability of Dedalus.
- Compared Dedalus's performance favorably to finite-volume codes.
- Successfully solved diverse complex PDE problems.

## Abstract

Numerical solutions of partial differential equations enable a broad range of scientific research. The Dedalus Project is a flexible, open-source, parallelized computational framework for solving general partial differential equations using spectral methods. Dedalus translates plain-text strings describing partial differential equations into efficient solvers. This paper details the numerical method that enables this translation, describes the design and implementation of the codebase, and illustrates its capabilities with a variety of example problems. The numerical method is a first-order generalized tau formulation that discretizes equations into banded matrices. This method is implemented with an object-oriented design. Classes for spectral bases and domains manage the discretization and automatic parallel distribution of variables. Discretized fields and mathematical operators are symbolically manipulated with a basic computer algebra system. Initial value, boundary value, and eigenvalue problems are efficiently solved using high-performance linear algebra, transform, and parallel communication libraries. Custom analysis outputs can also be specified in plain text and stored in self-describing portable formats. The performance of the code is evaluated with a parallel scaling benchmark and a comparison to a finite-volume code. The features and flexibility of the codebase are illustrated by solving several examples: the nonlinear Schrodinger equation on a graph, a supersonic magnetohydrodynamic vortex, quasigeostrophic flow, Stokes flow in a cylindrical annulus, normal modes of a radiative atmosphere, and diamagnetic levitation. The Dedalus code and the example problems are available online at http://dedalus-project.org/.