TwoDG.jl

A high-performance Julia framework for solving 2D partial differential equations using advanced finite element methods.

Cool Visuals

Compressible flow through a channel with a bump computed with 2D Euler equations showing evolution of Mach number

Pressure coefficient of a potential flow solution (left) and convection-diffusion solution on an unstructured mesh with Hybridizable Discontinuous Galerkin (HDG) (right)

Natural convection in a differentially heated cavity at Ra = 10⁴ (incompressible nonhydrostatic Navier-Stokes equations with the Boussinesq approximation, HDG k = 3): temperature (left) and speed (right). The steady hot-wall Nusselt number matches the de Vahl Davis benchmark.

Verification of the HDG incompressible Navier-Stokes solver with the Kovasznay flow at Re = 20: optimal k+1 convergence of velocity, pressure, and velocity gradient, and k+2 superconvergence of the exactly divergence-free, H(div)-conforming postprocessed velocity u*

Overview

TwoDG.jl provides a unified implementation of three powerful discretization schemes for 2D PDEs:

• Continuous Galerkin (CG) - Traditional continuous finite elements
• Discontinuous Galerkin (DG) - High-order explicit time-stepping methods
• Hybridizable Discontinuous Galerkin (HDG) - Efficient implicit solvers with static condensation and parallel capabilities

Whether you're studying wave propagation, compressible flows, or convection-diffusion phenomena, TwoDG.jl offers the numerical tools to tackle these problems with arbitrary polynomial orders and sophisticated error analysis.

Key Features

• Multiple PDE Types: Poisson, convection-diffusion, wave equations, Euler equations, and the incompressible Navier-Stokes equations (with optional Boussinesq buoyancy)
• High-Order Accuracy: Arbitrary polynomial order support (p-refinement) with Koornwinder orthogonal basis
• Efficient HDG: Static condensation dramatically reduces system size compared to standard DG
• Parallel HDG Solver: Multi-threaded assembly and solving for large-scale problems
• Rich Mesh Support: Built-in generators for squares, circles, L-shapes, NACA airfoils, and more
• Comprehensive Examples: Convergence studies, wave scattering, channel flows, and transport problems
• Performance Optimized: Inline functions and careful pre-allocation for computational efficiency

What Can You Do With It?

• Run convergence studies to verify optimal rates across different polynomial orders
• Compare discretization methods (CG vs DG vs HDG) on the same problems
• Simulate wave scattering on complex geometries with absorption boundaries
• Solve compressible flow problems including shock waves in channels
• Solve incompressible flow problems (steady or time-dependent Navier-Stokes, natural convection with the Boussinesq approximation) with the HDG method of Nguyen, Peraire & Cockburn (JCP, 2011)
• Analyze convection-diffusion transport with various stabilization parameters
• Develop new numerical methods using the extensible master element framework

Quick Example

using TwoDG

# Generate a mesh for your domain
mesh = squaremesh_rsquare(1.0, 5)  # 1x1 square, 5 elements per side

# Solve a Poisson problem with HDG
# (Check out examples in src/Apps/ for complete scripts)

Getting Started

Explore the example scripts in src/Apps/ to see the solvers in action:

• runhdg_poisson.jl - Poisson equation convergence studies
• runwavescattering.jl - Wave scattering on circular domains
• runeulerchannel.jl - Compressible Euler equations with shocks
• runconvection.jl - Pure convection with DG explicit time-stepping
• runhdg_ns_kovasznay.jl - Steady incompressible Navier-Stokes verification (Kovasznay flow, optimal k+1 convergence)
• runhdg_ns_boussinesq.jl - Natural convection in a heated cavity (incompressible nonhydrostatic Navier-Stokes with the Boussinesq approximation, validated against the de Vahl Davis benchmark)

Perfect for researchers in numerical analysis, students learning finite element methods, or anyone needing a flexible high-order PDE solver in Julia.