Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

Abstract

In this piece of work, 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 u tt = a ( x , t ) u xx + α ( x , t ) u x - 2 η 2 ( x , t ) u , u tt = a ( x , y , t ) u xx + b ( x , y , t ) u yy + α ( x , y , t ) u x + β ( x , y , t ) u y - 2 η 2 ( x , y , t ) u , and u tt = a ( x , y , z , t ) u xx + b ( x , y , z , t ) u yy + c ( x , y , z , t ) u zz + α ( x , y , z , t ) u x + β ( x , y , z , t ) u y + γ ( x , y , z , t ) u z - 2 η 2 ( x , y , z , t ) u , 0 < x , y , z < 1 , t > 0 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.