Glossary

8.2 Burgers' equation and shock formation

4 min read•august 15, 2024

combines nonlinear advection and diffusion, modeling simplified fluid flow. It's a key player in nonlinear PDEs, showing how smooth initial conditions can lead to shocks - sudden jumps in the solution that challenge our usual math tools.

Studying Burgers' equation helps us grasp trickier nonlinear PDEs like Navier-Stokes. We'll see how characteristics, jump conditions, and numerical methods tackle the equation's quirks, paving the way for understanding more complex problems.

Burgers' Equation from Conservation Laws

Conservation Laws and Equation Derivation

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  • Conservation laws in fluid dynamics express preservation of mass, momentum, and energy in a system
  • Burgers' equation combines nonlinear advection and diffusion terms to model simplified fluid flow
  • Derivation starts with one-dimensional continuity equation and momentum equation
  • Nonlinear advection term represents transport of quantity u by itself
  • Diffusion term accounts for viscous effects in the fluid
  • Equation can be written in conservative and non-conservative forms with specific mathematical properties
  • obtained by neglecting diffusion term, resulting in purely hyperbolic equation

Mathematical Representation and Properties

  • General form of Burgers' equation: ∂u∂t+u∂u∂x=ν∂2u∂x2\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}
  • Conservative form: ∂u∂t+∂∂x(12u2)=ν∂2u∂x2\frac{\partial u}{\partial t} + \frac{\partial}{\partial x}\left(\frac{1}{2}u^2\right) = \nu \frac{\partial^2 u}{\partial x^2}
  • Non-conservative form: ∂u∂t+u∂u∂x=ν∂2u∂x2\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}
  • Inviscid Burgers' equation: ∂u∂t+u∂u∂x=0\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = 0
  • Equation exhibits nonlinear behavior leading to wave steepening and shock formation
  • Solutions can develop discontinuities even with smooth initial conditions
  • Burgers' equation serves as prototype for studying more complex nonlinear PDEs (Navier-Stokes equations)

Shock Wave Formation and Propagation

Method of Characteristics Analysis

  • analyzes formation and propagation of shock waves in Burgers' equation
  • Characteristic curves in x-t plane represent paths along which information propagates in solution
  • For inviscid Burgers' equation, characteristics are straight lines with slope determined by initial condition
  • Characteristic equations: dxdt=u,dudt=0\frac{dx}{dt} = u, \quad \frac{du}{dt} = 0
  • Shock formation occurs when characteristic curves intersect, leading to multi-valued solutions in inviscid case
  • Time of shock formation can be calculated by finding when characteristics first intersect
  • Example: For initial condition u(x,0)=sin⁡(x)u(x,0) = \sin(x), shock forms at time t=1t = 1

Shock Propagation and Jump Conditions

  • Rankine-Hugoniot jump condition determines speed of propagation of
  • Jump condition: s=12(uL+uR)s = \frac{1}{2}(u_L + u_R), where s is shock speed, u_L and u_R are left and right states
  • Entropy conditions (Lax ) select physically relevant weak solutions
  • Lax entropy condition: uL>s>uRu_L > s > u_R
  • produces smooth solutions approximating shock waves as viscosity approaches zero
  • Shock width in viscous case proportional to viscosity coefficient
  • Example: For piecewise constant initial data, shock speed given by average of left and right states

Numerical Solutions for Burgers' Equation

Finite Difference Methods

  • Finite difference methods discretize Burgers' equation for numerical solution
  • Upwind scheme: ujn+1=ujn−ΔtΔxujn(ujn−uj−1n)u_j^{n+1} = u_j^n - \frac{\Delta t}{\Delta x}u_j^n(u_j^n - u_{j-1}^n)
  • Lax-Friedrichs scheme: ujn+1=12(uj+1n+uj−1n)−Δt2Δx(f(uj+1n)−f(uj−1n))u_j^{n+1} = \frac{1}{2}(u_{j+1}^n + u_{j-1}^n) - \frac{\Delta t}{2\Delta x}(f(u_{j+1}^n) - f(u_{j-1}^n))
  • Lax-Wendroff scheme: ujn+1=ujn−Δt2Δx(f(uj+1n)−f(uj−1n))+Δt22Δx2(f′(ujn)(f(uj+1n)−f(uj−1n)))u_j^{n+1} = u_j^n - \frac{\Delta t}{2\Delta x}(f(u_{j+1}^n) - f(u_{j-1}^n)) + \frac{\Delta t^2}{2\Delta x^2}(f'(u_j^n)(f(u_{j+1}^n) - f(u_{j-1}^n)))
  • Courant-Friedrichs-Lewy (CFL) condition ensures stability in explicit time-stepping schemes
  • for Burgers' equation: ΔtΔxmax⁡∣u∣≤1\frac{\Delta t}{\Delta x}\max|u| \leq 1

Advanced Numerical Techniques

  • and flux-limiting schemes capture shocks accurately without spurious oscillations
  • Implicit methods (Crank-Nicolson scheme) solve viscous Burgers' equation with improved stability
  • (Fourier and Chebyshev expansions) provide high-order accuracy for smooth solutions
  • Adaptive mesh refinement techniques resolve shock waves efficiently
  • Method of lines approach separates spatial and temporal discretizations, allowing use of sophisticated ODE solvers
  • Example: TVD (Total Variation Diminishing) schemes prevent spurious oscillations near shocks

Physical Meaning of Shock Waves

Physical Interpretation of Shock Solutions

  • Shock waves in Burgers' equation represent rapid changes in fluid velocity or density over small spatial regions
  • Formation of shock waves related to nonlinear steepening of wave profiles in physical systems
  • Speed of shock wave determined by states on either side of discontinuity ()
  • Entropy condition ensures information flows into shock, consistent with second law of thermodynamics
  • Viscous effects in real fluids smooth out discontinuities, resulting in thin transition layers analogous to shock waves

Applications and Analogies

  • Inviscid Burgers' equation serves as simplified model for studying phenomena (traffic flow, flood waves)
  • Balance between nonlinear advection and diffusion in Burgers' equation analogous to competing effects in many physical systems
  • Traffic flow analogy: Shock waves represent sudden changes in traffic density (traffic jams)
  • Flood wave analogy: Shock formation corresponds to steepening of flood wave front
  • Acoustic waves in nonlinear media exhibit similar shock formation processes
  • Burgers' equation provides insights into more complex fluid dynamics problems (supersonic flow, combustion)

Key Terms to Review (20)

Burgers' equation: Burgers' equation is a fundamental partial differential equation that describes the motion of a viscous fluid and is often used to model various phenomena in fluid dynamics and traffic flow. It combines both nonlinear advection and diffusion, making it a key equation for studying shock formation, which occurs when wave propagation causes steep gradients in the solution. This equation plays an important role in understanding how shocks develop and propagate in different physical contexts.
CFL Condition: The CFL condition, or Courant-Friedrichs-Lewy condition, is a mathematical criterion that ensures the stability of numerical schemes for solving partial differential equations. It establishes a relationship between the time step size, spatial discretization, and the wave speed in the equation being solved. A valid CFL condition is crucial for avoiding numerical instabilities, particularly when dealing with hyperbolic equations like Burgers' equation, where shock formation can occur.
Continuous Shock: A continuous shock refers to a type of discontinuity in the solution of a partial differential equation, where the solution remains bounded and continuous, but its derivatives may not be. This concept is crucial in understanding how certain equations, like Burgers' equation, can exhibit abrupt changes in state while maintaining overall continuity in certain aspects. In essence, continuous shocks demonstrate the balance between stability and instability within fluid dynamics and other physical phenomena.
Dirichlet Boundary Condition: A Dirichlet boundary condition specifies the values of a function on a boundary of its domain. This type of boundary condition is crucial when solving partial differential equations, as it allows us to set fixed values at the boundaries, which can greatly influence the solution behavior in various physical and mathematical contexts.
Entropy condition: The entropy condition is a mathematical criterion used to select physically relevant weak solutions to hyperbolic conservation laws, particularly in the presence of discontinuities like shocks. This concept helps to distinguish between multiple solutions by ensuring that the selected solution maintains certain physical properties, such as stability and monotonicity, which are essential when dealing with shock formation in equations like Burgers' equation.
Existence: In mathematics, existence refers to the condition where a solution to a given problem or equation can be found. It involves showing that at least one solution exists within a specified set of conditions, and this concept is crucial when considering the behavior and characteristics of equations, particularly those that describe physical phenomena or systems.
Finite Difference Method: The finite difference method is a numerical technique used to approximate solutions to differential equations by discretizing them into finite difference equations. This method transforms continuous derivatives into discrete differences, enabling the analysis of various partial differential equations through grid-based approximations.
Finite Time Blow-Up: Finite time blow-up refers to a phenomenon in the study of differential equations where a solution becomes unbounded or undefined in a finite amount of time. This situation often arises in nonlinear equations, where certain initial conditions can lead to singularities, such as shocks, indicating that the solution cannot continue past a specific time point. Understanding finite time blow-up is crucial when analyzing the behavior of solutions to equations like Burgers' equation, particularly in contexts involving shock formation.
Fluid Dynamics: Fluid dynamics is the study of how fluids (liquids and gases) behave and interact with forces, including how they flow, how they exert pressure, and how they respond to external influences. This area of study is crucial for understanding various physical phenomena and has applications across multiple fields, including engineering, meteorology, and oceanography.
Godunov's Method: Godunov's Method is a numerical technique used to solve hyperbolic partial differential equations, particularly in fluid dynamics and traffic flow. It is known for its ability to handle discontinuities, such as shock waves, by using a piecewise constant approximation of the solution and implementing Riemann problems at cell interfaces. This method is crucial when dealing with equations like Burgers' equation, which models shock formation in one-dimensional flows.
Initial Value Problem: An initial value problem (IVP) is a type of mathematical problem where one seeks to find a function that satisfies a differential equation along with specified values of that function at a given point in time or space. This concept is crucial as it establishes the conditions necessary for the existence and uniqueness of solutions to differential equations, allowing for accurate modeling in various fields.
Inviscid Burgers' Equation: The inviscid Burgers' equation is a fundamental nonlinear partial differential equation given by the form $$u_t + u u_x = 0$$, where $u$ represents the fluid velocity and subscripts denote partial derivatives. This equation describes the evolution of a wave-like motion in a one-dimensional space and is significant in understanding shock formation and propagation in fluid dynamics. Its simplicity makes it a key model for studying nonlinear phenomena, particularly in contexts like traffic flow and gas dynamics.
Method of characteristics: The method of characteristics is a technique used to solve certain types of partial differential equations (PDEs), particularly first-order PDEs, by transforming the PDE into a set of ordinary differential equations along characteristic curves. This approach allows for tracking the evolution of solutions over time, making it especially useful in contexts where shock formation and discontinuities are present.
Rankine-Hugoniot Condition: The Rankine-Hugoniot condition is a mathematical condition that describes the relationship between the speed of a discontinuity, such as a shock wave, and the values of a conserved quantity across that discontinuity. It plays a crucial role in understanding how solutions to hyperbolic partial differential equations, like Burgers' equation, can develop discontinuities known as shocks during their evolution.
Shock Wave: A shock wave is a type of propagating disturbance that moves faster than the speed of sound in a medium, creating a sharp change in pressure, temperature, and density. These waves are often formed in high-speed flows, such as those seen in supersonic flight or during explosive events, and they result in sudden changes that can drastically affect the surrounding environment. The understanding of shock waves is crucial for analyzing and predicting behaviors in fluid dynamics and related fields.
Spectral methods: Spectral methods are a class of numerical techniques used to solve differential equations by transforming the problem into a spectral space, often utilizing orthogonal functions like Fourier series or polynomials. These methods are highly effective for problems with smooth solutions and can provide very accurate results, particularly when analyzing complex phenomena like shock formation and stability in fluid dynamics.
Traffic flow model: A traffic flow model is a mathematical representation used to describe and analyze the movement of vehicles on roadways. It helps in understanding how traffic behaves under various conditions, including congestion, speed changes, and flow rates. These models are essential in deriving solutions to optimize traffic management and design transportation systems effectively.
Uniqueness: Uniqueness in the context of partial differential equations refers to the property that a solution to a given problem is the only one that satisfies both the equation and the specified conditions, such as initial and boundary conditions. This concept ensures that for well-defined mathematical problems, there’s no ambiguity in the solution, which is crucial for physical interpretations and applications.
Viscous Burgers' Equation: The viscous Burgers' equation is a fundamental partial differential equation that describes the motion of viscous fluid flow and combines elements of both diffusion and non-linear convection. This equation captures the behavior of shock waves and the transition from smooth to discontinuous solutions in fluid dynamics, making it an important model for understanding shock formation in various physical contexts.
Weak solution: A weak solution is a generalized concept of a solution to a partial differential equation that may not be differentiable but satisfies the equation in an integral sense. This approach allows for the inclusion of solutions that exhibit discontinuities, such as shock waves, particularly in conservation laws. Weak solutions are crucial in the study of equations like Burgers' equation, as they help address issues like the formation of shocks and the breakdown of classical solutions.
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