The physicists and mathematicians have put a lot of eﬀorts in the numerical analysis of various types of partial diﬀerential equations on unbounded domain. The time- dependent partial diﬀerential equations(PDEs) also have a wide range of applications in physics, geography and many other interdisciplines. This thesis is concerned with the numerical solutions of such kind of partial diﬀerential equations on unbounded spatial domain, especially the Korteweg-de Vries(KdV) equations. Since it is unable to solve the problem directly due to its unboundedness, the common way to surpass such diﬃculty is to introduce proper conditions on the truncated artiﬁcial boundaries and to approximate the problem on a bounded domain, which is also known as the Absorbing Boundary Conditions(ABCs).One of the main contributions of this thesis is to design accurate local absorbing boundary conditions for linearized KdV equations and to extend the method to non- linear KdV equations on unbounded domain. Pad´e approximation is the main tool to approximate the cubic root in the construction of local absorbing boundary conditions(LABCs) for a linearized KdV equation on unbounded domain. Besides, we also introduce the continued fraction method in the approximation of cubic root. To avoid the high-order derivatives in the absorbing boundary conditions, a sequence of auxiliary variables are applied accordingly. Then the original problem on unbounded domain is reduced to an approximated initial boundary value(IBV) problem deﬁned on a ﬁnite domain.Based on previous work, we are able to extend the method to the design of eﬃcient local absorbing boundary conditions for nonlinear KdV equations on unbounded domain. The unifying approach method is applied to this nonlinear case. The idea of the unifying approach method is to separate inward- and outward-going waves and to build suitable approximated linear operator with a “one-way operator”. Then we unite the approximated linear operator with the nonlinear subproblem and propose boundary conditions for the nonlinear subproblem along the artiﬁcial boundaries.The numerical simulations are given to demonstrate the eﬀectiveness and accuracy of our local absorbing boundary conditions.Keywords: Korteweg-de Vries equation; Local absorbing boundary conditions; Pad´e approximation; Continued fraction method; Unifying approach.
|Date of Award||1 Sept 2014|
|Supervisor||Xiaonan WU (Supervisor)|
- Differential equations
- Korteweg-de Vries equation
- Numerical analysis