Numerical methods for hyperbolic conservation laws
Introduction
I will start by giving a general introduction to scalar conservation laws \(\partial_t u + \partial_x f(u) = 0\). Existence, uniqueness and stability of entropy solutions.
- S. Mishra, U. S. Fjordholm and R. Abgrall. Numerical methods for conservation laws and related equations. (Chapter 1-3)
- H. Holden and N. H. Risebro. Front tracking for hyperbolic conservation laws. (Chapter 2)
Monotone schemes
The classical approach to proving convergence of numerical methods for conservation laws. Uses a rather standard compactness approach to show convergence.
- M. G. Crandall and A. Majda. Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980), pp. 1-21.
- S. Mishra, U. S. Fjordholm and R. Abgrall. Numerical methods for conservation laws and related equations. (Chapter 4)
- H. Holden and N. H. Risebro. Front tracking for hyperbolic conservation laws. (Section 3.1)
Second-order TVD schemes
To increase the order of accuracy of finite volume schemes, a common approach is to perform "reconstruction", a type of interpolation of the piecewise constant data. If done correctly, this leads to second-order accurate, TVD, L^infty bounded schemes.
- S. Mishra, U. S. Fjordholm and R. Abgrall. Numerical methods for conservation laws and related equations. (Chapter 5)
The Discontinuous Galerkin method
DG methods are finite element methods where the solution is allowed to be discontinuous across element edges. Cockburn and Shu introduced these methods for conservation laws in a series of papers (1988-1990). The convergence proof takes a similar approach as for monotone schemes.
Compensated compactness
Compensated compactness is a powerful method for showing convergence of approximate solutions of PDEs, originally due to F. Murat and L. Tartar (1978-1979). The main result is the div-curl lemma. We give a rather elementary proof of the div-curl lemma which uses properties of the solution of Poisson's equation. Time permitting, we show how this method can be used to prove convergence of solutions of the regularized problem \(\partial_t u + \partial_x f(u) = \varepsilon \partial_{xx}u\) as \(\varepsilon\to0\).
- L. C. Evans. Weak Convergence Methods for Nonlinear Partial Differential Equations. (Chapter 5B)
- G.-Q. Chen. Compactness Methods and Nonlinear Hyperbolic Conservation Laws. (Chapter 2)
- C. M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. (Chapter XVII)
Spectral methods
A spectral method expands the solution of a PDE as a Fourier series, truncates the series and derives an effective equation for this approximate solution. E. Tadmor (1989) proved that this approximation scheme converges spectrally. The proof of convergence uses basic Fourier analysis and compensated compactness.
Streamline diffusion finite element methods
The streamline diffusion (SD) method is a finite element method (FEM) which adds numerical diffusion in a judicious manner, namely in the direction of the flow. If done correctly, this stabilizes the method sufficiently for convergence, while retaining a high order of accuracy. The development of SD-FEM methods for conservation laws is due to Johnson and Szepessy (1987-1990). Their proof of convergence uses compensated compactness.
Nonlocal kinetic equations
"Kinetic" equations of the form \(\partial_t\rho + \nabla\cdot(\rho V[\rho]) = 0\), where \(\rho\) is a probability density and \(V[\rho]\) is a nonlocal (and possibly nonlinear) operator, appear in many different applications such as the modelling of traffic, tumor growth and animal swarms. The classical existence proof uses a particle approximation. We will consider a simple finite volume method and prove convergence to a measure-valued solution. The proof uses some tools from probability theory but is otherwise similar to the proof of convergence of monotone schemes for conservation laws.
The Navier-Stokes equations
The Navier-Stokes equation \(\partial_t u + \nabla\cdot(u\otimes u) + \nabla p = \nu\Delta u,\ \nabla\cdot u = 0\) for the evolution of a viscous, incompressible fluid has a long history. The original proof of existence of weak solutions, due to Leray (1934), is actually a convergence proof for a Galerkin (or "finite element") approximation. We will study (a modern take on) this proof, recalling some of the necessary functional-analytical details along the way. Time permitting, we will also sketch a proof that Leray solutions are unique in two, but not three, spatial dimensions. (This last issue is closely related to the Clay Millennium Problem.)