Part I: Mathematical Theory

§ 0. Introduction, Examples

0.1 Hyperbolic conservation laws in 1D

• Definition 1: hyperbolic

0.2 The derivation of conservation laws

• Example 1: 1D gas flow in a tube

§ 1. Scalar conservation laws

1.1 Examples

• Example 1: Linear Advection Equation
• Example 2: Advection equation with variable coefficients
• Example 3: (viscous) Burgers equation

1.2 shock development, weak solutions

• Example 1: Shock development in Burgers equation

Definition 1: weak solution

1.3 The Riemann problem - shock waves - rarefaction waves

• Riemann Problem
• Rankine-Hugoniot jump condition
• Example 1: Burgers equation

Manipulation with conservation laws

1.4 The entropy condition

• Entropy condition (1st version)
• Entropy functions
• Entropy condition (2nd version)

1.5 Summary

1.6 Further examples of scalar equations

• Example 1: Traffic Flow Model

Example 2: Buckley-Leverett equation (two-phase flow)

Example 3: Kynch equation (sedimentation processes)

§ 2. Linear Hyperbolic Systems

• Definition 1: hyperbolic
• Definition 2: strictly hyperbolic

2.1 Examplee

• Example 1: Euler equations of gas dynamics, shock tube, airfoil flow
• Example 2: shallow-water equation
• Example 3: equations of magnetohydrodynamics

2.2 Characteristic variables

• Domain of dependence
• p-characteristics
• Simple waves
• Example 1: wave equation

2.3 Linearization of nonlinear systems

• Example 1: 1D Euler equations of gas dynamics, sound waves
• Example 2: isothermal Euler equations

2.4 The Riemann problem

• Riemann problem
• Rankine-Hugoniot jump condition for systems

2.5 The Hugoniot locus in the phase space

• Definition 1: Phase space, Hugoniot locus
• Example 1:

§ 3. Nonlinear hyperbolic systems

3.1 Hugoniot locus, shocks

• Riemannp roblem
• Definition 1: Hugoniot locus
• Hugoniot curves
• Theorem 1:
• Example 1: isothermal Euler equations
• Theorem 2: (local) unique solvability of the Riemann problem

3.2 The entropy condition of Lax

• entropy condition of Lax
• pure nonlinear field
• Example 1: isothermal Euler equations
• Linear degeneration
• contact discontinuity
• modified entropy condition

3.3 Rarefaction waves, Riemann invariants

• Example 1: isothermal Euler equations
• Calculation of rarefaction waves (1st method)
• General solution to the Riemann problem
• Example 1: isothermal Euler equations
• Theorem 1: (local) unique solvability of the entropy-fulfilling Riemann problem
• Calculation of contact discontinuities and rarefaction waves using Riemann invariants (2nd method)
• Definition 1: p-Riemann invariants
• Example 1: isothermal Euler equations

3.4 The Riemann problem for the full Euler equations

• isentropic Euler equations
• Riemann invariants
• Solution of the Riemann problem

Part II: Numerical Methods

§ 4. Numerical methods for linear equations

• Grid (points), approximation, hyperbolic mesh ratio
• Finite difference method
• Example 1: explicit Euler method
• Example 2: implicit Euler method
• Example 3: Lax-Friedrichs method
• Example 4: Leapfrog method
• Example 5: Lax-Wendroff method
• General explicit one-step methods Uⁿ⁺¹=H_k(Uⁿ)

4.1 Error, Consistency

• Error (function)
• Definition 1: Convergence
• Example 1: explicit Euler method
• Definition 2: local discretization error, consistency (order)
• Example 2: Lax-Friedrichs method

4.2 Stability

• Recursion formula for the error
• Definition 1: Stability
• Theorem 1: Lax-Richtmyer's equivalence theorem
• Example 1: Lax-Friedrichs method
• Definition 2: (numerical) domain of dependence
• CFL (Courant-Friedrichs-Levy) condition
• Example 2: one-sided method
• Example 3: 3-point scheme
• Example 1: Lax-Friedrichs Procedure (continued)

Upwind methods

• Example 2: one-sided method
• Example 4: Upwind method

4.3 Calculation of discontinuous solutions

• Example 1: u_t+u_x=0, methods 1./2. Order, numerical results
• Modified equations (for 1st order methods)
• Example 2: Lax-Friedrichs method
• artificial diffusion
• Example 3: Upwind method
• Error for discontinuous data
• Modified equations (for 2nd order methods)
• Example 4: Lax-Wendroff method
• Example 5: Beam Warming scheme
• Dispersion relation
• Phase velocity, dispersion
• group velocity

§ 5. Conservative methods for nonlinear equations

5.1 The Lax-Wendroff Theorem

• Example 1: Example 1: Upwind method for u_t+u u_x=0, numerical Results
• Definition 1: conservative, numerical flow, consistent
• Interpretation
• Example 2: (conservative) upwind method for u_t+(u²/2)_x=0
• Example 3:Lax-Friedrichs method for u_t+f(u)_x=0
• Example 4: Lax-Wendroff generalizations for u_t+f(u)_x=0
• Example 5: Two-step method
  • Richtmyer-two step Lax-Wendroff method
  • MacCormack method

Definition 2: Total variation

Definition 3: Convergence

Theorem 1: Theorem of Lax-Wendroff

Entropy condition

• Example 6: Riemann problem for u_t+(u²/2)_x=0, generalized upwind
• transonic rarefaction waves

Approximation of the entropy solution

• Discrete entropy condition
• Numerical entropy flow
• Theorem 2: Limiting solution is the entropy solution

5.2 The Godunov Scheme

• Example 1: Upwind, Lax-Friedrichs for u_t+au_x=0
• Example 2: CIR (Courant-Isaacson-Rees) method
• Godunov method
• Calculation of the entropy solution
• Theorem 1: Godunov method satisfies discrete entropy condition
• Corollary 1: Limiting solution of the Godunov method is entropy solution

5.3 Approximate solution of the Riemann problem

• approximative Riemann solver
• 2 approaches for approximate Godunov method
• Theorem 1: Conservativity and consistency of the approximate Godunov method
• The method of Roe
• Definition 1: Roe matrix
• Advantages and disadvantages of the Roe method
• Construction of a "sonic entropy fix"
• scalar problems
• Example 1:
• Example 2:
• Systems
• Example 3:
• Example 4:
• Construction of a Roe matrix
• Example 5: Roe matrix for isothermal Euler equations

§ 6. Convergence of scalar methods (nonlinear stability)

6.1 TV-Stability

• concept of convergence
• Definition 1: TV-Stability
• Theorem 1: criterion for TV stability
• Theorem 2: Convergence Theorem

6.2 Stability criteria

• Theorem 3: TVD property of Entropy solutions
• Definition 2: TVD (total variation diminishing) method
• Theorem 4: monotonicity preservation property of entropy solutions
• Definition 3: monotonicity preserving

Theorem 5: TVD methods are monotonicity preserving

Theorem 6: L¹-contraction property of entropy solutions

Definition 4: l¹-contracting

Theorem 7: l¹-contracting methods are TVD

Example 1: left-side Upwind method

Theorem 8: Monotonicity of Entropy Solutions

Korollar 1: Minimum-Maximum Property of Entropy Solutions

Definition 5: monotone

Example 2: Lax-Friedrichs method

Theorem 9: monotone schemes are l¹-contracting

Theorem 10: monotone methods have at most Consistency Order 1

Theorem 11: Limiting solution of consistent, monotone methods is entropy solution

§ 7. High-order methods (construction of high-resolution TVD methods)

7.1 Artificial viscosity

• Example 1: Lax-Wendroff for u_t+au_x=0 with Artificial viscosity
• Theorem 1: Linear monotonicity preserving methods have at most consistency order 1
• Corollary 1: Stability concepts from Chapter 6.2 are equivalent for linear methods
• solution-dependent artificial viscosity

7.2 Methods with flux limitation (flux-limiter)

• Example 2: F_L: Upwind flux, F_H:Lax-Wendroff flux for u_t+au_x=0
• Theorem 2: TVD methods close to extreme points not 2nd order
• 2nd order resolution
• Theorem 3: TVD criterion for general method with flux limitation
• ``Superbee limiter`` of Roe
• Flux limiter of van Leer

Generalization for linear systems

7.3 Methods with slope limitation (slope-limiter)

• Algorithm: reconstruction, evolution, projection
• Theorem 4: TVD criterion for general method with slope limitation
• Example 3: Interpretation of Lax-Wendroff for u_t+au_x=0
• minmod-slope
• Generalization for linear systems
• Generalization for nonlinear systems

§ 8. Semi-discrete Methods

• method of lines (MOL)
• Evolution equations for cell averages
• improved approximation of the flux
• Example 1: piecewise linear approximation
• Reconstruction by primitive functions
• ENO (essentially nonoscillatory) method
• Definition 1: ENO-method of order p

§ 9. Multidimensional problems

• Finite Volume Methods
• Semi-discrete Methods
• Splitting Schemes
• Example 1: scalar Advection equation in 2D
• Splitting of order 1
• Strang-Splitting
• Example 2: Linear conservation law in 2D
• TVD-Methods in 2D
• Theorem 1: TVD-Methoden in 2D have almost consistency order 1

Part III: Excursion

§ 10. Initial boundary value problems (IBVPs) for conservation laws

10.1 Boundary conditions for linear Systems

• Example 1: linear System in 1D
• Example 2: 1D Euler equations
• sub/supersonic inflow boundary
• Outflow extrapolation
• Example 3: No-flux Boundary conditions in R²
• Example 4: Absorbing Boundary conditions
• Definition 1: strong solution
• Theorem 1: Existence / uniqueness of strong solution of IBVP in 2D
• Definition 2: well-posed IBVP
• Theorem 2: ill-posed IBVP
• Theorem 3: Necessary condition for well-posedness
• Lemma 1: Uniform Kreiss condition

10.2 Boundary Conditions for Nonlinear Scalar Equations

• Definition 1: BV(Omega)
• Theorem 1: Compactness in L¹(Omega)
• Lemma 1: Existence of a trace mapping
• Theorem 2: Convergence of the viscosity solution
• Motivation of the Boundary Conditions
• Definition 2: Weak solution of the IBVP
• Theorem 3: Existence / uniqueness of weak solution of IBVP

§ 11. A kinetic Approach for hyperbolic Systems

• Example 1: Advection equation
• Example 2: Burgers equation
• Kinetic schemes