Question

Method of characteristics to solve mass momentum energy equations

Answer 1

Burgers' equation is or, equivalently, . The wave speed depends on the solution, . That is, the speed of a point on the solution profile will depend on the vertical coordinate u of the point. Inviscid Burgers' equation is not of the form of the linear first order PDE (1), as it is nonlinear, so our earlier analysis do not apply directly. However, from our experience with the constant coefficient (4) and variable coefficient (5) advection equations, we are led to set the characteristic equation to be . If x(t) is a solution of this equation, then u(x(t),t) is the restriction of u to this curve. Also, along this curve,



Thus, this solution u(x(t),t) will not change with time along the curve, so it is a characteristic curve. If we know the initial condition , we can find the characteristic curve by substituting this value into the characteristic equation . The right hand side of the equation is constant, indicating that the characteristic curves will be straight lines, as in the constant coefficient advection equation case. Specifically, the characteristic curve is . The solution of the initial value problem can be written as . The solution is given implicitly and, in all but the simplest cases, it is impossible to determine the solution explicitly. The characteristics are straight lines, but the lines do not all have the same slope, so it is possible for the characteristics to intersect. If we write the