DG1D_ADVECTION
Discontinuous Galerkin Solution of 1D Advection Equation

DG1D_ADVECTION is a MATLAB library which uses the Discontinuous Galerkin Method (DG) to approximate a solution of the unsteady 1D advection equation. The original version of the code was written by Jan Hesthaven and Tim Warburton.

A 1D version of the advection equation has the form

        du/dt + 2 pi du/dx = 0  for 0 < x < 2
 
        u(0,t) = - sin(2*pi*t)
        u(x,0) = sin(x)

Licensing:

      Permission to use this software for noncommercial
      research and educational purposes is hereby granted
      without fee.  Redistribution, sale, or incorporation
      of this software into a commercial product is prohibited.

      THE AUTHORS OR PUBLISHER DISCLAIMS ANY AND ALL WARRANTIES
      WITH REGARD TO THIS SOFTWARE, INCLUDING ALL IMPLIED
      WARRANTIES OF MERCHANTABILITY AND FITNESS FOR ANY
      PARTICULAR PURPOSE.  IN NO EVENT SHALL THE AUTHORS OR
      THE PUBLISHER BE LIABLE FOR ANY SPECIAL, INDIRECT OR
      CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER
      RESULTING FROM LOSS OF USE, DATA OR PROFITS.

Languages:

DG1D_ADVECTION is available in a MATLAB version.

Related Data and Programs:

BVP4C, MATLAB programs which illustrate how to use the MATLAB command bvp4c(), which can solve boundary value problems (BVP's) in one spatial dimension.

dg1d_advection_test

DG1D_BURGERS, a MATLAB library which uses the Discontinuous Galerkin Method (DG) to approximate a solution of the unsteady 1D Burgers Equation. The original version of the code was written by Jan Hesthaven and Tim Warburton.

DG1D_HEAT, a MATLAB library which uses the Discontinuous Galerkin Method (DG) to approximate a solution of the unsteady 1D heat Equation. The original version of the code was written by Jan Hesthaven and Tim Warburton.

DG1D_MAXWELL, a MATLAB library which uses the Discontinuous Galerkin Method (DG) to approximate a solution of Maxwell's equations. The original version of the code was written by Jan Hesthaven and Tim Warburton.

DG1D_POISSON, a MATLAB program which applies the discontinuous Galerkin method (DG) to a 1D version of the Poisson equation, by Beatrice Riviere.

FD1D_ADVECTION_DIFFUSION_STEADY, a MATLAB program which applies the finite difference method to solve the steady advection diffusion equation v*ux-k*uxx=0 in one spatial dimension, with constant velocity v and diffusivity k.

FD1D_ADVECTION_FTCS, a MATLAB program which applies the finite difference method to solve the time-dependent advection equation ut = - c * ux in one spatial dimension, with a constant velocity, using the FTCS method, forward time difference, centered space difference.

FD1D_ADVECTION_LAX_WENDROFF, a MATLAB program which applies the finite difference method to solve the time-dependent advection equation ut = - c * ux in one spatial dimension, with a constant velocity, using the Lax-Wendroff method to treat the time derivative.

FD1D_BURGERS_LAX, a MATLAB program which applies the finite difference method and the Lax-Wendroff method to solve the non-viscous time-dependent Burgers equation in one spatial dimension.

FD1D_BURGERS_LEAP, a MATLAB program which applies the finite difference method and the leapfrog approach to solve the non-viscous time-dependent Burgers equation in one spatial dimension.

FD1D_BVP, a MATLAB program which applies the finite difference method to a two point boundary value problem in one spatial dimension.

FD1D_HEAT_EXPLICIT, a MATLAB program which uses the finite difference method and explicit time stepping to solve the time dependent heat equation in 1D.

FD1D_HEAT_IMPLICIT, a MATLAB program which uses the finite difference method and implicit time stepping to solve the time dependent heat equation in 1D.

FD1D_HEAT_STEADY, a MATLAB program which uses the finite difference method to solve the steady (time independent) heat equation in 1D.

FD1D_PREDATOR_PREY, a MATLAB program which implements a finite difference algorithm for predator-prey system with spatial variation in 1D.

FD1D_WAVE, a MATLAB program which applies the finite difference method to solve the time-dependent wave equation in one spatial dimension.

FEM1D, a MATLAB program which applies the finite element method to a linear two point boundary value problem in a 1D region.

FEM1D_BVP_LINEAR, a MATLAB program which applies the finite element method, with piecewise linear elements, to a two point boundary value problem in one spatial dimension.

Author:

The original MATLAB source code was written by Jan Hesthaven and Tim Warburton.

Reference:

Jan Hesthaven, Tim Warburton,
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications,
Springer, 2007,
ISBN: 978-0387720654.

Source Code:

Advec1D.m, integrates the 1D advection equation over a time interval, given an initial condition.
AdvecRHS1D.m, evaluate the right hand sides flux in the advection equation.
BuildMaps1D.m, sets up connectivity and boundary tables for the nodes.
Connect1D.m, builds the global connectivity arrays.
Dmatrix1D.m, initializes the differentiation matrices.
GeometricFactors1D.m, computes metric elements for the local mappings.
Globals1D.m, declares global variables.
GradJacobiP.m, evaluates derivatives of the Jacobi polynomial.
GradVandermonde1D.m, initialializes the gradient of the modal basis.
JacobiGL.m, compute the Nth order Gauss-Lobatto quadrature points.
JacobiGQ.m, compute the Nth order Gauss quadrature points.
JacobiP.m, evaluate the Jacobi polynomial.
Lift1D.m, compute the surface integral term for the Discontinuous Galerkin formulation.
MeshGen1D.m, generate a uniform spatial grid.
Normals1D.m, compute the outward normals at element faces.
StartUp1D.m, sets building operators, grid, metric and connectivity.
timestamp.m, prints the YMDHMS date as a timestamp.
Vandermonde1D.m, initialize the 1D Vandermonde matrix.

Last modified on 04 January 2019.