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Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms. (English) Zbl 1122.65381
Summary: We introduce a new idea and obtain stability interval for explicit difference schemes of $O(k^2+h^2)$ for one, two and three space dimensional second-order hyperbolic equations $$\align u_{tt}&= a(x,t)u_{xx}+ \alpha(x,t)u_x- 2\eta^2(x,t)u,\\ u_{tt}&=a(x,y,t)u_{xx}+ b(x,y,t)u_{yy}+ \alpha(x,y,t)u_x+ \beta(x,y,t)u_y- 2\eta^2(x,y,t)u,\endalign$$ and $$\align u_{tt}=&a(x,y,z,t)u_{xx}+ b(x,y,z,t)u_{yy}+ c(x,y,z,t)u_{zz}+\alpha(x,y,z,t)u_x+\\ +&\beta(x,y,z,t)u_y+\gamma(x,y,z,t)u_z- 2\eta^2(x,y,z,t)u, \quad 0<x,y,z<1,\ t>0, \endalign$$ subject to appropriate initial and Dirichlet boundary conditions, where $h>0$ and $k>0$ are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of $O(k^2)$ in order to obtain numerical solution of $u$ at first time step in a different manner.

MSC:
65M06Finite difference methods (IVP of PDE)
35L15Second order hyperbolic equations, initial value problems
65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
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References:
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