where q = q(x, t) is a function of one spatial dimension and time, and u is a velocity that is constant in space and time. This is called an advection equation<\/strong>, as it describes the time dependent shifting of the function q(x) along x with a velocity u. The solution at any time t > t0<\/sub> can be described as a function of the state at time t0: q(x, t) = q(x \u2212 ut, 0)<\/em>.<\/p>\n\n\n\n This is a so-called initial value problem in which the state at any time t > t0 can be uniquely found when the state at time t = t0<\/sub> is fully given. The characteristics of this problem are straight lines: xchar<\/sub>(t) = x(0)<\/sup>char<\/sub> + ut. This is a family of lines in the (x, t) plane, each of which is labeled by its own unique value of (x(0)<\/sup>char<\/sub>).<\/p>\n\n\n\n Let us consider a set of linear equations that can be written in the form:\u2202t<\/sub>Q + A\u2202x<\/sub>Q = 0, where Q is a vector of m components and A is an m \u00d7 m matrix. This system is called hyperbolic if the matrix A is diagonalizable with real eigenvalues. The matrix is diagonalizable if there exists a complete set of eigenvectors ei<\/sub>, i.e. if any vector can be written as: <\/p>\n\n\n\n In this case one can write<\/p>\n\n\n\n We can define a matrix in which each column is one of the eigenvectors:<\/p>\n\n\n\n Then we can transform the first equation into:<\/p>\n\n\n\n which with Q\u02dc = R\u22121<\/sup>Q then becomes:<\/p>\n\n\n\n where A\u02dc = diag (\u03bb1, \u00b7\u00b7\u00b7 , \u03bbm). Not all \u03bbi must be different from each other. This system of equations has in principle m sets of characteristics. But any set of characteristics that has the same characteristic velocity as another set is usually called the same set of characteristics. So in the case of 5 eigenvalues, of which three are identical, one typically says that there are three sets of characteristics.<\/p>\n<\/div><\/div>\n\n\n\n Let us focus on the general conservation equation:<\/p>\n\n\n\n where, as ever, Q = (q1<\/sub>, \u00b7\u00b7\u00b7 , qm<\/sub>) and F = (f1<\/sub>, \u00b7\u00b7\u00b7 , fm<\/sub>). In general, F is not always a linear function of Q, i.e. it cannot always be formulated as a matrix A times the vector Q (except if A is allowed to also depend on Q, but then the usefulness of writing F = AQ is a bit gone). So let us assume that F is some non-linear function of Q. Let us, for the moment, assume that F = F(Q, x) = F(Q), i.e. we assume that there is no explicit dependence of F on x, except through Q. Then we get<\/p>\n\n\n\n where \u2202F\/\u2202Q is the Jacobian matrix, which depends, in the non-linear case, on Q itself. We can nevertheless decompose this matrix in eigenvectors (which depend on Q) and we obtain<\/p>\n\n\n\n Here the eigenvalues \u03bb1, \u00b7\u00b7\u00b7 , \u03bbm and eigenvectors (and hence the meaning of Q\u02dc) depends on Q. In principle this is not a problem. The characteristics are now simply given by the state vector Q itself. The state is, so to speak, self-propagating. We are now getting into the kind of hyperbolic equations like the hydrodynamics equations, which are also non-linear self-propagating.<\/p>\n<\/div><\/div>\n\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" The hydrodynamics equations are nothing more than signal-propagation equations, and equations of this kind are called hyperbolic equations. The equations of hydrodynamics are only a member of the more general class of hyperbolic equations, and there are many more examples of hyperbolic equations than just the equations of hydrodynamics. The simplest form of a hyperbolic equation Consider the equation \u2202tq + u\u2202xq = 0. where q = q(x, t) is a function of one spatial dimension and time, and u is a velocity that is constant in space and time. This is called an advection equation, as it describes the time dependent shifting of the function q(x) along x with a velocity u. The solution at any time t > t0 can be described as a function of the state at time t0: q(x, t) = q(x \u2212 ut, 0). This is a so-called initial value problem in which the state at any time t > t0 can be uniquely found when the state at time t = t0 is fully given. The characteristics of this problem are straight lines: xchar(t) = x(0)char + ut. This is a family of lines in the (x, t) plane, each of which is labeled by its own unique value of (x(0)char).<\/p>\n","protected":false},"author":1,"featured_media":1750,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false,"footnotes":""},"class_list":["post-1747","page","type-page","status-publish","has-post-thumbnail","hentry"],"ams_acf":[],"yoast_head":"\n![\"\"](\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/10\/image-11.png?resize=960%2C446&ssl=1\")
Hyperbolic sets of equations: the linear case with constant Jacobian<\/h2>\n\n\n\n
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<\/figure><\/div>\n\n\n\nHyperbolic equations: the non-linear case<\/h2>\n\n\n\n
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