ConservationLawsDiffEq

Collection of explicit numerical schemes for solving systems of Conservations Laws (finite volume methods), using method of Lines and an ODE Solver.

Each scheme returns a semidiscretization (discretization in space) that represents a ODE system. Time integration is performed then using OrdinaryDiffEq.

The general conservation laws problem is represented by the following PDE,

Solutions follow a conservative finite difference (finite volume) pattern. This method updates cell averages of the solution u. For a particular cell i it has the general form

Where the numerical flux is an approximate solution of the Riemann problem at the cell interface .

Features

Mesh:

At the moment only a Cartesian 1D uniform mesh is available, using Uniform1DFVMesh(N,[a,b]) command. Where

N = Number of cells

a,b = start and end coordinates.

The semidiscretization is obtained by using:

getSemiDiscretization(f,scheme,mesh,[boundary_conditions]; Df, use_threads,numvars)

where f is the flux function defined as a julia function.

scheme is the explicit finite volumes scheme used to discretize the flux (see next section).

mesh a valid finite volumes mesh.

boundary_conditions a set of boundary conditions among: Dirichlet(), ZeroFlux() and Periodic(). Note: Dirichlet boundary values are defined by the initial condition.

Df: is an optional Jacobian of the flux function.

use_threads: true or false.

numvars: number of variables of the Conservation Laws system.

Schemes

The explicit numerical schemes in space currently available are the following:

Lax-Friedrichs method

(LaxFriedrichsScheme()), Global/Local L-F Scheme (GlobalLaxFriedrichsScheme(), LocalLaxFriedrichsScheme()), Second order Law-Wendroff Scheme (LaxWendroffScheme()), Ritchmeyer Two-step Lax-Wendroff Method (LaxWendroff2sScheme())

• R. LeVeque. Finite Volume Methods for Hyperbolic Problems.Cambridge University Press. New York 2002

TECNO Schemes

(FVTecnoScheme(Nflux;ve, order))

• U. Fjordholm, S. Mishra, E. Tadmor, Arbitrarly high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. 2012. SIAM. vol. 50. No 2. pp. 544-573

High-Resolution Central Schemes

(FVSKTScheme(;slopeLimiter=GeneralizedMinmodLimiter()))

Kurganov, Tadmor, New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection–Diffusion Equations, Journal of Computational Physics, Vol 160, issue 1, 1 May 2000, Pages 241-282

Second-Order upwind central scheme

(FVCUScheme(;slopeLimiter=GeneralizedMinmodLimiter()))

• Kurganov A., Noelle S., Petrova G., Semidiscrete Central-Upwind schemes for hyperbolic Conservation Laws and Hamilton-Jacobi Equations. SIAM. Sci Comput, Vol 23, No 3m pp 707-740. 2001

Dissipation Reduced Central upwind Scheme:

Second-Order (FVDRCUScheme(;slopeLimiter=GeneralizedMinmodLimiter())), fifth-order (FVDRCU5Scheme(;slopeLimiter=GeneralizedMinmodLimiter()))

• Kurganov A., Lin C., On the reduction of Numerical Dissipation in Central-Upwind # Schemes, Commun. Comput. Phys. Vol 2. No. 1, pp 141-163, Feb 2007.

Component Wise Weighted Essentially Non-Oscilaroty (WENO-LF)

(FVCompWENOScheme(;order))

• C.-W. Shu, High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Review, 51:82-126, (2009).

Component Wise Mapped WENO Scheme

(FVCompMWENOScheme(;order))

• A. Henrick, T. Aslam, J. Powers, Mapped weighted essentially non-oscillatory schemes: Achiving optimal order near critical points. Journal of Computational Physics. Vol 207. 2005. Pages 542-567

Characteristic Wise WENO (Spectral) Scheme

(FVSpecMWENOScheme(;order))

• R. Bürger, R. Donat, P. Mulet, C. Vega, On the implementation of WENO schemes for a class of polydisperse sedimentation models. Journal of Computational Physics, Volume 230, Issue 6, 20 March 2011, Pages 2322-2344

Note: OrdinaryDiffEq callbacks can be used in order to fix a CFL constant value, or recover the dt from adaptative ODE methods in the cases when the finite volumes scheme needs its value (getCFLCallback and get_adaptative_callback methods, see examples for more information about its use)

Note: Limiters available: GeneralizedMinmodLimiter(;θ=1.0), MinmodLimiter(), OsherLimiter(;β=1.0), SuperbeeLimiter().

Example

Scalar Burgers equation:

# u(x,t)_t+(0.5*u²(x,t))_{x}=0
# u(0,x) = f0(x)

using ConservationLawsDiffEq
using OrdinaryDiffEq
using LinearAlgebra

const CFL = 0.5
# First define the problem data (Jacobian is optional but useful)
Jf(u) = u           #Jacobian
f(u) = u^2/2        #Flux function
f0(x) = sin(2*π*x)  #Initial data distribution

# Now discretize the domain
mesh = Uniform1DFVMesh(10, [0.0, 1.0])

# Now get a explicit semidiscretization (discrete in space) du_h(t)/dt = f_h(u_h(t))
f_h = getSemiDiscretization(f,LaxFriedrichsScheme(),mesh,[Periodic()]; Df = Jf, use_threads = false,numvars = 1)

#Compute discrete initial data
u0 = getInitialState(mesh,f0,use_threads = true)

#Setup ODE problem for a time interval = [0.0,1.0]
ode_prob = ODEProblem(f_h,u0,(0.0,1.0))

#Setup callback in order to fix CFL constant value
cb = getCFLCallback(f_h, CFL)

#Estimate an initial dt
dt = update_dt!(u0, f_h, CFL)

#Solve problem using OrdinaryDiffEq
sol = solve(ode_prob,SSPRK22(); dt = dt, callback = cb)

#Plot solution
#Wrap solution so we can plot using dispatch
u_h = fv_solution(sol, mesh)
using Plots
plot(u_h,tidx = 1,lab="uo",line=(:dot,2)) #Plot initial data
plot!(u_h,lab="LF")                       #Plot LaxFriedrichsScheme solution

Disclamer

** developed for personal use, some of the schemes have not been tested enough!!!**