Pallavika, V.K.Kalyani1, S.K.Chakraborty and Amalendu Sinha1 - IGU

Pallavika et al.… (10)Instead of using the wave equation, which is asecond order hyperbolic system, the elastodynamicequations are given by,… (11)The above equations can be transformed into thefollowing first-order hyperbolic system:...… (12)Stability AnalysisWhen the finite difference approximation is used tosolve the equation of motion of seismic wavepropagation in the time domain, the stability criteriamust be satisfied and the finite difference schemeapplied must be convergent and stable. By stability,one means that errors made at one stage ofcalculations do not cause increasingly large errors asthe computations progress, but rather error dies out.Also by convergent one means that the results of themethod approach the analytical results as Δt and Δxapproach zero. Many authors (Cao & Greenhalgh1998, Mitchell 1969) have developed stability fordifferent finite difference approximations The stabilitycondition obtained for the finite difference scheme(13), using the initial error in α, β and ν.at t = 0, isfound to bewhere,..(14)The central finite-difference scheme developed in(10) is used to discretize the derivatives of thesegoverning equations. Thus, the explicit numericalscheme, equivalent to the system (12), is given by. .. .. .… (13)where l is the index for time discretization, m for x-axis discretization, and n for z-axis discretization. Δtis the grid step in time, Δx and Δz are the grid stepsfor the x-axis and for the z-axis, respectively.Initial and Boundary ConditionsUnder the initial conditions both stress and velocityare zero everywhere in the medium at time t = 0 tomake the medium in equilibrium (Virieux 1986).Boundary conditions considered here are the Neumanncondition (stress-free conditions) and the Dirichletcondition (zero velocity or zero displacementconditions).For the finite difference scheme (13) to be stableshould be always less than or equal to 1.Consequently, ω will be real, and the error will notgrow with time. Thus, the condition for stability isto make the time-mesh interval Δt smaller than orequal to , where is the local wavevelocity for Δx = Δz.The stability condition (14) also gives the relationbetween frequency and wave number k to be satisfiedby the solution. The phase and group velocitiesdetermined by (14) approach the correct local valuesif Δt, Δx and Δz are small. Also, for small Δt, Δx andΔz one can approximate Sinθ by θ and obtain, from(14),… (15)It is important to know that how small Δx andΔz should be so that the above relation isapproximately correct. The limit depends on thewavelength ,. Rewriting (14), the phase velocityis given by..(16)168