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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.

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)
Full Text: DOI
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