Riemann invariants are mathematical transformations made on a a system of quasi-linear

Linear differential equation

Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...

 first order partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s to make them more easily solvable. Riemann invariants are constant along the characteristic curves

Method of characteristics

In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation...

 of the partial differential equations where they obtain the name invariant

Invariant (mathematics)

In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

. They were first obtained by Bernhard Riemann

Bernhard Riemann

Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

 in his work on plane waves in gas dynamics.

Mathematical theory

Consider the set of hyperbolic partial differential equations of the form

where and are the elements of the matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

  and where and are elements of vector

Vector

Vector, a Latin word meaning "carrier", may refer in English to:-In computer science:*A one-dimensional array**Vector , a data type in the C++ Standard Template Library...

s. It will be asked if it is possible to rewrite this equation to

To do this curves will be introduced in the plane defined by the vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

 . The term in the brackets will be rewritten in terms of a total derivative

Total derivative

In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.The total derivative of a function f, of several variables, e.g., t, x, y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative...

 where are parametrized as

comparing the last two equations we find

which can be now written in characteristic form

where we must have the conditions,

where can be eliminated to give the necessary condition

so for a nontrival solution is the determinant

For Riemann invariants we are concerned with the case when the matrix is an identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

 to form

notice this is homogeneous due to the vector being zero. In characteristic form the system is with

Where is the left eigenvector of the matrix and is the characteristic speeds of the eigenvalues of the matrix which satisfy

To simplify these characteristic equations we can make the transformations such that

which form

An integrating factor

Integrating factor

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...

  can be multiplied in to help integrate this. So the system now has the characteristic form on

which is equivalent to the diagonal system

Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...

The solution of this system can be given by the generalized hodograph method.

Example

Consider the shallow water equations

Shallow water equations

The shallow water equations are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid .The equations are derived from depth-integrating the Navier–Stokes...

write this system in matrix form

where the matrix from the analysis above the eigenvalues and eigenvectors need to be found.The eigenvalues are found to satisfy

to give

and the eigenvectors are found to be

where the riemann invariants are

In shallow water equations there is the relation to give the riemann invariants

to give the equations

Which can be solved by the hodograph transformation. If the matrix form of the system of pde's is in the form

Then it may be possible to multiply across by the inverse matrix so long as the matrix determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

of is not zero.