applicable to tidal flow, flows in lakes and reservoirs, river flow, and wave 

 motion where the wavelength is significantly greater than the water depth 

 (hence the terminology "long-wave equations"). 



Numerical Solution Method 



33. Equations 2 and 5 constitute a system of first-order nonlinear 

 partial differential equations of the hyperbolic type. These equations do not 

 have analytical solutions except for certain special cases. 



34. Numerical methods for the solution of the equations of unsteady 

 flow have been known since the time of Massau (1889). These solution methods 

 may be classified as either direct or characteristic. In the direct method, 

 the finite-difference representation is based directly on the primary equa- 

 tions. In the characteristic method, the equations are first transformed into 

 the characteristic form, and this form is then used to develop the finite - 

 difference representation. In the direct method, a fixed mesh of points on 

 the time-space plane is commonly employed to identify grid points, that is to 

 say, times and locations at which solutions are to be obtained. In the 

 characteristic method, solutions may be obtained at the intersection of the 

 characteristic curves on the time -space plane or at fixed points of a rectan- 

 gular mesh by interpolation. 



35. Finite-difference numerical solution schemes used in the direct and 

 characteristic methods may be further classified as being either explicit or 

 implicit. In explicit solution methods, the finite-difference equations are 

 usually reduced to linear algebraic equations by some form of approximation 

 from which the unknowns can be individually isolated explicitly, i.e., 

 evaluated directly. In implicit methods, the finite-difference equations are 

 generally expressed as nonlinear algebraic equations from which the unknowns 

 cannot be isolated. Depending on the manner in which derivatives are replaced 

 by finite differences, whether forward, centered, or backward, a variety of 

 numerical methods can be developed. 



36. The fixed-mesh explicit method is the first and well-known numeri- 

 cal method for the solution of the equations of unsteady flow. It was 

 developed by Stoker (1957) and colleagues and applied to river flow problems. 

 The method is subject to a stringent stability condition that imposes a 



15 



