### U.W. Bangor - School of Informatics - Mathematics Preprints 1996

# Computational Applied Mathematics

#### 96.01 : GARDNER, L.R.T, GARDNER, G.A. & DOGAN, A.

### A least squares finite element scheme for Burger's equation

#### Abstract:

Burger's equation is solved by a least squares technique using linear space-time finite elements. Standard problems are solved to assess the properties of the algorithm.#### Published in:

#### 96.02 : GARDNER, L.R.T, GARDNER, G.A. & DOGAN, A.

### A least squares finite element scheme for the RLW equation

#### Abstract:

The RLW equation is solved by a least squares technique using linear space-time finite elements.In simulations of the migration of a single solitary wave this algorithm is shown to have higher accuracy and better conservation than a recent difference scheme based on cubic spline interpolation functions. In addition, for very small amplitude waves (<= 0,09) it has a higher accuracy than an approach using quadratic B-spline finite elements within Galerkin's method.

The development of an undular bore is modelled.

#### Published in:

*Communications in Numerical Methods in Engineering*12 (1996) 795-804.

#### 96.03 : GARDNER, L.R.T, GARDNER, G.A. & DOGAN, A.

### A Galerkin finite element scheme for the RLW equation

#### Abstract:

The RLW equation is solved by Galerkin's method using linear space finite elements.In simulations of the migration of a single solitary wave this algorithm is shown to have good accuracy for small amplitude waves. In addition, for very small amplitude waves (<= 0.09) it has higher accuracy than an approach using quadratic B-spline finite elements within Galerkin's method.

The interaction of two solitary waves is modelled for small amplitude waves.

#### Published in:

#### 96.06 : GARDNER, L.R.T, GARDNER, G.A. & DOGAN, A.

### A Petrov-Galerkin finite element scheme for Burgers' equation

#### Abstract:

Burgers' equation is solved by a Petrov-Galerkin method using quadratic B-spline spacial finite elements. A linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is obtained via a Crank-Nicolson approach involving a product approximation. Standard problems are solved to assess the properties of the algorithm.#### Published in:

*Arabian J for Sci. and Eng.*22 (1997) 100-109.

#### 96.09 : GARDNER, L.R.T, GARDNER, G.A. & DOGAN, A.

### A Petrov-Galerkin algorithm for the RLW equation

#### Abstract:

The regularised long wave equation is solved by a Petrov-Galerkin method using quadratic B-spline spatial finite elements. A linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is obtained via a Crank-Nicolson approach involving a product approximation. The motion of solitary waves is studied to assess the properties of the algorithm. The development of an undular bore is studied.#### Published in:

#### 96.18 : GARDNER, L.R.T, GARDNER, G.A. AYOUB, F.A. & AMEIN, N.K.

### Modelling an undular bore with B-splines

#### Abstract:

The development of an undular bore is governed by the regularised long wave equation. A Petrov-Galerkin B-spline finite element method for its solution is set up. The numerical model is validated by showing that it models well the motion of a single solitary wave. The development of an undular bore is studied through computer simulation.#### Published in:

*Comput. Methods Appl. Mech. Engrg.*147 (1997) 147-152.

#### 96.19 : GARDNER, L.R.T, GARDNER, G.A. AYOUB, F.A. & AMEIN, N.K.

### Simulations of the EW undular bore

#### Abstract:

The equal width equation is solved by a Petrov-Galerkin method using quadratic B-spline spatial finite elements. A linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is obtained via a Crank-Nicolson approach involving a product approximation. The motion of solitary waves is studied to assess the properties of the algorithm. The development of an EW undular bore is investigated and compared with that of the RLW bore.#### Published in:

*Commun. Numer. Meth. Engng.*13 (1997) 583-592.