Contractive difference schemes for symmetric hyperbolic systems

Authors:
Philip Brenner and Vidar Thomée

Journal:
Math. Comp. **25** (1971), 205-217

MSC:
Primary 65N05

DOI:
https://doi.org/10.1090/S0025-5718-1971-0297151-1

MathSciNet review:
0297151

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider the initial-value problem for a constant-coefficient symmetric hyperbolic system with initial-values vanishing in a half-space. Consider also a finite difference operator consistent with the system. Conditions are given in terms of the orders of dissipation and accuracy which ensure that the solution of the discrete problem tends to zero exponentially with the mesh-width in half-spaces where the solution of the continuous problem vanishes.

**[1]**Mats Y. T. Apelkrans,*On difference schemes for hyperbolic equations with discontinuous initial values*, Math. Comp.**22**(1968), 525–539. MR**0233527**, https://doi.org/10.1090/S0025-5718-1968-0233527-6**[2]**Philip Brenner and Vidar Thomée,*Stability and convergence rates in 𝐿_{𝑝} for certain difference schemes*, Math. Scand.**27**(1970), 5–23. MR**0278549**, https://doi.org/10.7146/math.scand.a-10983**[3]**Philip Brenner and Vidar Thomée,*Estimates near discontinuities for some difference schemes*, Math. Scand.**28**(1971), 329–340 (1972). MR**0305613**, https://doi.org/10.7146/math.scand.a-11028**[4]**S. Gerschgorin, "Über die Abgrenzung der Eigenwerte einer Matrix,"*Izv. Akad. Nauk SSSR*, v. 7, 1931, pp. 749-754.**[5]**G. W. Hedstrom,*Norms of powers of absolutely convergent Fourier series*, Michigan Math. J.**13**(1966), 393–416. MR**0203340****[6]**G. W. Hedstrom,*Norms of powers of absolutely convergent Fourier series in several variables*, Michigan Math. J.**14**(1967), 493–495. MR**0222549****[7]**G. W. Hedstrom,*The rate of convergence of some difference schemes*, SIAM J. Numer. Anal.**5**(1968), 363–406. MR**0230489**, https://doi.org/10.1137/0705031**[8]**Heinz-Otto Kreiss,*Über Matrizen die beschränkte Halbgruppen erzeugen*, Math. Scand.**7**(1959), 71–80 (German). MR**0110952**, https://doi.org/10.7146/math.scand.a-10563**[9]**Heinz-Otto Kreiss,*On difference approximations of the dissipative type for hyperbolic differential equations*, Comm. Pure Appl. Math.**17**(1964), 335–353. MR**0166937**, https://doi.org/10.1002/cpa.3160170306**[10]**Heinz-Otto Kreiss and Einar Lundqvist,*On difference approximations with wrong boundary values*, Math. Comp.**22**(1968), 1–12. MR**0228193**, https://doi.org/10.1090/S0025-5718-1968-0228193-X**[11]**P. D. Lax,*Differential equations, difference equations and matrix theory*, Comm. Pure Appl. Math.**11**(1958), 175–194. MR**0098110**, https://doi.org/10.1002/cpa.3160110203**[12]**Stanley Osher,*On systems of difference equations with wrong boundary conditions*, Math. Comp.**23**(1969), 567–572. MR**0247785**, https://doi.org/10.1090/S0025-5718-1969-0247785-6**[13]**Beresford Parlett,*Accuracy and dissipation in difference schemes*, Comm. Pure Appl. Math.**19**(1966), 111–123. MR**0196957**, https://doi.org/10.1002/cpa.3160190109**[14]**Vidar Thomée,*Parabolic difference operators*, Math. Scand.**19**(1966), 77–107. MR**0209693**, https://doi.org/10.7146/math.scand.a-10797**[15]**Vidar Thomée,*Stability theory for partial difference operators*, SIAM Rev.**11**(1969), 152–195. MR**0250505**, https://doi.org/10.1137/1011033

Retrieve articles in *Mathematics of Computation*
with MSC:
65N05

Retrieve articles in all journals with MSC: 65N05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0297151-1

Keywords:
Symmetric hyperbolic system,
initial-value problem,
difference scheme,
contractive,
dissipative,
stability,
accuracy,
Fourier transform

Article copyright:
© Copyright 1971
American Mathematical Society