## An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions.(English)Zbl 0995.65093

Summary: An unconditionally stable alternating direction implicit (ADI) method of $$O(k^2+ h^2)$$ of M. Lees type [J. Soc. Ind. Appl. Math. 10, 610-616 (1962; Zbl 0111.29204)] for solving the three space dimensional linear hyperbolic equation $$u_{tt}+ 2\alpha u_t+ \beta^2u= u_{xx}+ u_{yy}+ u_{zz}+ f(x,y,z,t)$$, $$0< x, y$$, $$z< 1$$, $$t> 0$$ subject to appropriate initial and Dirichlet boundary conditions is proposed, where $$\alpha> 0$$ and $$\beta\geq 0$$ are real numbers. For this method, we use a single computational cell. The resulting system of algebraic equations is solved by a three step split method. The new method is demonstrated by a suitable numerical example.

### MSC:

 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L05 Wave equation 65F10 Iterative numerical methods for linear systems

Zbl 0111.29204
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### References:

 [1] DOI: 10.1080/00207169508804400 · Zbl 0845.65046 [2] DOI: 10.1137/0110046 · Zbl 0111.29204 [3] DOI: 10.1093/imamat/11.1.105 · Zbl 0259.65085 [4] DOI: 10.1137/0706006 · Zbl 0175.16203 [5] Jain M. K., ”Numerical solution of differential equations”, 2. ed. (1984) · Zbl 0536.65004
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