1.1 Partial Differential Equations-Some Basics
5
The evolution of the solution is particularly simple when C = 0 and B / A is some
constant value c, in which case (LI) reduces to
au au
-+c-=O.
at ax
(1.4)
If u(x. 0) = f tx), the solution to the preceding is f(x - ct), implying that
the initial perturbations in u translate without distortion at a uniform velocity c.
Equation (1.4). which is often referred to as the one-way wave equation or the
constant-wind-speed advection equation, is the simplest mathematical model for
wave propagation. Although it is quite simple. (1.4) is a very useful prototype
problem for testing numerical methods because solutions to more complex linear hyperbolic systems can often be expressed as the superposition of individual
waves govemed by one-way wave equations.
A system of partial differential equations in two independent variables is hyperbolic if it has a complete set of characteristic curves that can in principle be
used to locally determine the solution from appropriately prescribed initial data.
As a first example, consider a constant-coefficient linear system of the form
Bu;
aus _ 0
-a + LJars- a - .
t
s=1
x
r = 1.2• . . . • n.
(1.5)
This system may be altematively written as
where uppcrcase boldface letters represent matrices and lowercase boldface letters
denote vectors. The system is hyperbolic if there exist bounded matrices T and
T- 1 such that T- 1 AT = D. where D is a diagonal matrix with real eigenvalues
dii- When the system is hyperbolic, it can be transformed to
(1.6)
by defining v = T- 1 u. Since D is a diagonal matrix, each element vj of the vector of unknown functions may be determined by solving a simpler scalar equation of the form (1.4). Each diagonal element of D is associated with a family
of characteristic curves along which the perturbations in Vi propagate at speed
dx jdt = djj . The wave-like character of the solution can be demonstrated by
Fourier transforming (1.6) with respect to x to obtain
av
-
at
+ikDv = O.
(1.7)
where vis the Fourier transform of v and k is the wave number, or dual variable. In order to satisfy (1.7). the jth component of v must be a wave of the form
5
The evolution of the solution is particularly simple when C = 0 and B / A is some
constant value c, in which case (LI) reduces to
au au
-+c-=O.
at ax
(1.4)
If u(x. 0) = f tx), the solution to the preceding is f(x - ct), implying that
the initial perturbations in u translate without distortion at a uniform velocity c.
Equation (1.4). which is often referred to as the one-way wave equation or the
constant-wind-speed advection equation, is the simplest mathematical model for
wave propagation. Although it is quite simple. (1.4) is a very useful prototype
problem for testing numerical methods because solutions to more complex linear hyperbolic systems can often be expressed as the superposition of individual
waves govemed by one-way wave equations.
A system of partial differential equations in two independent variables is hyperbolic if it has a complete set of characteristic curves that can in principle be
used to locally determine the solution from appropriately prescribed initial data.
As a first example, consider a constant-coefficient linear system of the form
Bu;
aus _ 0
-a + LJars- a - .
t
s=1
x
r = 1.2• . . . • n.
(1.5)
This system may be altematively written as
where uppcrcase boldface letters represent matrices and lowercase boldface letters
denote vectors. The system is hyperbolic if there exist bounded matrices T and
T- 1 such that T- 1 AT = D. where D is a diagonal matrix with real eigenvalues
dii- When the system is hyperbolic, it can be transformed to
(1.6)
by defining v = T- 1 u. Since D is a diagonal matrix, each element vj of the vector of unknown functions may be determined by solving a simpler scalar equation of the form (1.4). Each diagonal element of D is associated with a family
of characteristic curves along which the perturbations in Vi propagate at speed
dx jdt = djj . The wave-like character of the solution can be demonstrated by
Fourier transforming (1.6) with respect to x to obtain
av
-
at
+ikDv = O.
(1.7)
where vis the Fourier transform of v and k is the wave number, or dual variable. In order to satisfy (1.7). the jth component of v must be a wave of the form