View source for Method of characteristics

A method for the numerical integration of equations of hyperbolic type. In the domain of hyperbolicity there is a linear combination of initial equations in which there occur only interior derivatives along characteristic surfaces. Then the equations to be solved simplify substantially. In the method of characteristics the solution is computed on a characteristic grid, which is constructed in the process of calculation, and so the domain of dependency of the solution can be determined exactly. For the method of characteristics the existence of a solution and its convergence have been proved. The widest application of the method of characteristics is in the solution of problems of the mechanics of continuous media (see [[#References|[1]]]). For example, the equations in characteristic form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m0635801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m0635802.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

are a linear combination of the traditional equations of gas dynamics: continuity, momentum and energy. Here and below <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m0635803.png" /> is the density, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m0635804.png" /> is the velocity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m0635805.png" /> is the intrinsic energy of a unit mass, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m0635806.png" /> is the pressure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m0635807.png" /> is the temperature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m0635808.png" /> is the spatial coordinate, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m0635809.png" /> is the time. The Cauchy problem asks for a solution in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358010.png" /> for given data on the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358011.png" />. Entropy is the name for an integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358012.png" /> of the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358013.png" /></td> </tr></table>

Then (1) has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>

On the left-hand sides of (1prm) and (2) there stand the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358017.png" />, taken in the directions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

the so-called characteristics. The system (1), (2) has three families of real characteristics. Along the characteristics (3) the relation

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holds, and along the characteristic (4) the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/m063580a.gif" />

Figure: m063580a

hold. Through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358022.png" /> (see Fig.) there passes a characteristic (4) on the side of increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358023.png" />,

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Through a point close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358025.png" /> on the right, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358026.png" />, there passes a characteristic (4) of another family

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358028.png" /> is the point of intersection of the characteristics. When one replaces the differential relations (5), which hold along characteristics, by differences, one obtains the algebraic system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358029.png" /></td> </tr></table>

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from which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358032.png" /> can be determined. From <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358033.png" /> one draws the characteristic (3),

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up to the intersection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358035.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358036.png" />. The value of the entropy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358037.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358038.png" /> is determined by means of interpolation between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358040.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358041.png" />). From the equations

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one can find the values of the intrinsic energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358044.png" /> and the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358045.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358046.png" />. When the data at the two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358048.png" /> are known, one can find the solution at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358049.png" /> for large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358050.png" />. This procedure of computations is repeated for every pair of points. Then, by using new points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358051.png" /> instead of the original <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358053.png" />, one takes the next step in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358054.png" />. The calculation is made for the required values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063580/m06358055.png" />. However, since the equations of gas dynamics are non-linear, the calculation may come to a stop at a certain moment of time if the characteristics of one family touch each other or intersect.

The difference scheme thus described is of the first order of accuracy (an analogue of Euler's method of polygonal lines for solving ordinary differential equations). An increase in accuracy can be achieved by repetitions of the calculations, etc.

Using the method of characteristics one can solve stationary multi-dimensional problems in a domain of hyperbolicity (for gas dynamics — of supersonic flow). One can also determine the position of secondary shock waves at places where the characteristics of a single family intersect or touch. By the method of characteristics one can tackle only problems with a small number of discontinuities, since under an accumulation of singularities the computations become tedious. A computation by the method of characteristics consists of a number of elementary problems: the calculation of an interior point, of a point on the shock wave, or on a body around which the flow is taking place, etc.

One can construct numerical schemes for the method of characteristics, which make it possible to carry out the calculations in "layers" , the grid-characteristic method (see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> , ''An attempt at calculating plane and axially-symmetric supersonic flows of a gas by the method of characteristics'' , Moscow (1961) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> K.M. Magomedov, A.S. Kholodov, "The constructions of difference schemes for hyperbolic equations based on characteristic relations" ''USSR Math. Math. Comp.'' , '''9''' : 2 (1969) pp. 158–176 ''Zh. Vyshch. Mat. i Mat. Fiz.'' , '''9''' : 2 (1969) pp. 373–386</TD></TR></table>



====Comments====
The method of characteristics goes back to J. Massau (see [[#References|[a2]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W.F. Ames, "Numerical methods for partial differential equations" , Acad. Press (1977) {{MR|0519123}} {{ZBL|0577.65077}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Massau, "Mémoire sur l'intégration graphique des équations aux derivées partielles" , F. Mayer-van Loo , Ghent (1899) {{MR|0052203}} {{ZBL|30.0312.01}} </TD></TR></table>