The Characteristic Initial Value Problem for the Wave Equation and Riemann's Method

1954 ◽  
Vol 61 (10) ◽  
pp. 702 ◽  
Author(s):  
M. H. Protter
Keyword(s):  
Wave Equation ◽  
Initial Value Problem ◽  
Initial Value ◽  
Characteristic Initial Value Problem

Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations

1990 ◽  
Vol 427 (1872) ◽  
pp. 221-239 ◽  
Keyword(s):  
Cauchy Problem ◽  
Perfect Fluid ◽  
Initial Value Problem ◽  
Einstein Equations ◽  
Initial Value ◽  
Fluid Source ◽  
Null Hypersurfaces ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Characteristic Initial Value Problem

A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.

Numerical relativity. II. Numerical methods for the characteristic initial value problem and the evolution of the vacuum field equations for space‒times with two Killing vectors

1983 ◽  
Vol 386 (1791) ◽  
pp. 373-391 ◽  
Keyword(s):  
General Relativity ◽  
Numerical Methods ◽  
Plane Wave ◽  
Numerical Solution ◽  
Initial Value Problem ◽  
Vacuum Field ◽  
Field Equations ◽  
Initial Value ◽  
Wave Space ◽  
Characteristic Initial Value Problem

This is the second of a sequence of papers on the numerical solution of the characteristic initial value problem in general relativity. Although the equations to be integrated have regular coefficients, the nonlinearity leads to the occurrence of singularities after a finite evolution time. In this paper we first discuss some novel techniques for integrating the equations right up to the singularities. The second half of the paper presents as examples the numerical evolution of the Schwarzschild and certain colliding plane wave space‒times.

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