The Finite-difference Method for Seismologists: An by JOZEF KRISTEK, AND LADISLAV HALADA PETER MOCZO

By JOZEF KRISTEK, AND LADISLAV HALADA PETER MOCZO

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2000). Von Neumann‘s Analysis of Stability The way we investigated the stability of the FD schemes in the previous section is acceptable but does not give a possibility to quantify a measure of (in)stability. This is possible with the von Neumann’s method. Because the stability is the key concept we will illustrate the von Neumann’s method on three examples: a. The unstable velocity-stress FD scheme on a conventional grid given by equations (17) and (18), b. The stable velocity-stress FD scheme on a conventional grid given by equations (21) and (22), c.

2 2004 Moczo, Kristek, Halada: The Finite-Difference Method for Seismologists. An Introduction Table 1. Coefficients of the absorbing boundary conditions (107) Reynolds A 00 Clayton & Engquist (A1) 1977 0 A01 0 0 A 02 0 0 cΔt h cΔ t h cΔt h cΔ t 1+ h A12 0 0 A 20 0 0 A 21 0 −1 − A 22 0 A10 A11 50 1− 1978 0 1− cΔ t h cΔ t h Emerman & Stephen 1983 0 Δt − h c Δt + h c Liu & Archuleta 2000 0 0 2h c Δt + h c 2h c Δt + h c 0 h1x c 1t + h1t c xt + h xt − h1x c 1t 0 Δt − h c Δt + h c −h1x c xt −1 −h xt c 1t − h1t c xt 0 − h xt c xt −h1t c 1t SPICE – Marie Curie Research Training Network SPICE Research and Training Workshop, Venice, Italy, Sep.

CI Δ 2 i S VIm Define vector U mI ⎡V m ⎤ ⎢ I ⎥ ⎢ m−1 ⎥ ⎢V ⎥ m UI = ⎢ I ⎥ ⎢T m ⎥ ⎢ I ⎥ ⎢ m−1 ⎥ ⎣⎢TI ⎦⎥ (46) and matrix G ⎡ ⎢ ⎢ ⎢ G = ⎢⎢ ⎢C ⎢ I ⎢ ⎣⎢ 0 1 Δ 2i S 0 1 1 Δ 2i S ρI 0 0 0 0 0 1 ⎤ 0⎥ ⎥ ⎥ 0⎥ . ⎥ 1⎥⎥ 0⎥⎦⎥ (47) Then system of equations can be written in the matrix form U mI +1 = G U mI . (48) Similarly to the case of the equation (38) for the FDE (30) we get the characteristic equation C 1 − 2Λ2 + 4Δ2 I S 2 Λ2 + Λ4 = 0 . (49) ρI Denoting a = Δ2 and 26 CI 2 S ρI (50) SPICE – Marie Curie Research Training Network SPICE Research and Training Workshop, Venice, Italy, Sep.

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