Compact Diference Schemes For Several Types Of Diferential Equations And Explicit STS Acceleration

In the frst part of the paper, we extend the second-order nonsymmetricscheme to third-and fourth-order nonsymmetric schemes by linear representa-tion using low-order diferential for high-order diferential in the local truncationerrors based on the second-order nonsymmetric scheme of third-order boundaryproblem. The bandwidth of the matrix obtained from extend schemes does notincrease except on the boundary. Furthermore, the bandwidth of the matrixof the fourth-order scheme is narrower than that of the previous second orderscheme. Convergence analysis proved that our schemes are convergent and theaccuracy is third and fourth order respectively. Numerical experiments validatethe accuracy and efciency of our schemes. Meanwhile, Other boundary condi-tions also are considered in this part.In the second part, we construct three diferent high-order implicit compactschemes for the KdV-Burgers equation. The fourth-order nonsymmetric schemeis constructed by the ideas mentioned in the frst part. We use three-level schemefor the time discretization and then it does not require special treatment forthe nonlinear term (such as extrapolation). Numerical experiments verifed theaccuracy and efciency of our schemes.In the third part, we investigate the performance of two classes of STS(Super-time-stepping) schemes which conjugating with the frst order explicitEuler method and high order explicit Runge-Kutta methods for time discretiza-tion referred here as the frst order STS scheme and high order STS schemesrespectively. Numerical experiments shows that accelerating efect of the frstorder STS scheme for heat equation, nonlinear Stefan problem and viscous non-linear Burgers equation is apparent compared with standard explicit schemes.But for the high order STS schemes, accelerating efect is not apparent. Weplot the absolute stability region of frst order STS scheme and high order STSschemes respectively through theoretical analysis, and we fnd that the abso- lute stability region of high order STS schemes is smaller than standard explicitRunge-Kutta schemes both on real axis and imaginary axis. Our results showthat simply extend the STS method to Runge-Kutta schemes is not established.