Solving diffusion equation using wavelet method

A wavelet-based numerical method based on wavelets of Hermite cubic splines is presented for computing singularly perturbed convection-dominated diffusion equation. The advantages of the method are explained. To improve the accuracy of singular areas, wavelets are configured hierarchically for solving algebraic equations. Numerical examples show that the proposed method has good efficiency and precision.     

Construction of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines

The paper presents a simple procedure for the construction of quasi-interpolation operators in spaces of m-harmonic splines in R^d, which reproduce polynomials of high degree. The procedure starts from a generator ?₀, which is easy to derive but with corresponding quasi-interpolation operator reproducing only linear polynomials, and recursively defines generators ?₁,?₂,…,?_m−1 with corresponding quasi-interpolation operators reproducing polynomials of degree up to 3,5,…,2m−1 respectively. The construction of ?_j from ?_j−1 is explicit, simple and independent of m. The special case d=1 and the special cases d=2,m=2,3,4 are discussed in details.     

Approximation of boundary values in the boundary element method

Various techniques may be applied to the approximation of the unknown boundary functions involved in the boundary element method (BEM). Several techniques have been examined numerically to find the most efficient. Techniques considered were: Lagrangian polynomials of the zeroth, first and second orders; spline functions; and the novel weighted minimization technique used successfully in the finite difference method (FDM) for arbitrarily irregular meshes. All these approaches have been used in the BEM for the numerical analysis of plates with various boundary conditions.Both coarse and fine grids on the boundary have been assumed. Maximal errors of the deflections of each plate and the bending moments have been found and the effective computer CPU times determined.Analysis of the results showed that, for the same computer time, the greatest accuracy was obtained by the weighted FDM approach. In the case of the Lagrange approximation, higher order polynomials have proved more efficient. The spline technique yielded more accurate results, but with a higher CPU time.Two discretization approaches have been investigated: the least-squares technique and the collocation method. Despite the fact that the simultaneous algebraic equations obtained were not symmetric, the collocation approach has been confirmed as clearly superior to the least-squares technique, because of the amount of computation time used.     

A hierarchical basis preconditioner for the biharmonic equation on the sphere

^** Email: jan.maes{at}cs.kuleuven.be In this paper, we propose a natural way to extend a bivariatePowell–Sabin (PS) B-spline basis on a planar polygonaldomain to a PS B-spline basis defined on a subset of the unitsphere in [graphic: see PDF] . The spherical basis inherits many properties of the bivariatebasis such as local support, the partition of unity propertyand stability. This allows us to construct a C¹ continuous hierarchicalbasis on the sphere that is suitable for preconditioning fourth-orderelliptic problems on the sphere. We show that the stiffnessmatrix relative to this hierarchical basis has a logarithmicallygrowing condition number, which is a suboptimal result comparedto standard multigrid methods. Nevertheless, this is a hugeimprovement over solving the discretized system without preconditioning,and its extreme simplicity contributes to its attractiveness.Furthermore, we briefly describe a way to stabilize the hierarchicalbasis with the aid of the lifting scheme. This yields a waveletbasis on the sphere for which we find a uniformly well-conditionedand (quasi-) sparse stiffness matrix.     

Modified nodal cubic spline collocation for elliptic equations

We consider the modified nodal cubic spline collocation method for a general, variable coefficient, second order partial differential equation in the unit square with the solution subject to the homogeneous Dirichlet boundary conditions. The bicubic spline approximate solution satisfies both the Dirichlet boundary conditions and a perturbed partial differential equation at the nodes of a uniform partition of the square. We prove existence and uniqueness of the approximate solution and derive an optimal fourth order maximum norm error bound. The resulting linear system is solved efficiently by a preconditioned iterative method. Numerical results confirm the expected convergence rates. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Quartic B‐spline collocation algorithms for numerical solution of the RLW equation

The collocation method based on quartic B‐spline interpolation is studied for numerical solution of the regularized long wave (RLW) equation. The time‐split RLW equation is also solved with the quartic B‐spline collocation method. Numerical accuracy is tested by obtaining the single solitary wave solution. Then, interaction, undulation and evolution of solitary waves are studied. Solutions are compared with available results. Conservation quantities are computed for all test experiments. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

PIECEWISE SEMIALGEBRAIC SETS

Semialgebraic sets are objects which are truly a special feature of real algebraic geometry. This paper presents the piecewise semialgebraic set, which is the subset of Rn satisfying a boolean combination of multivariate spline equations and inequalities with real coefficients. Moreover, the stability under projection and the dimension of C^μ piecewise semialgebraic sets are also discussed.     

Interpolating clothoid splines with curvature continuity

An efficient algorithm for the computation of a C² interpolating clothoid spline is herein presented. The spline is obtained following an optimisation process, subject to continuity constraints. Among the 9 various targets/problems considered, there are boundary conditions, minimum length path, minimum jerk, and minimum curvature (energy). Some of these problems are solved with just a couple of Newton iterations, whereas the more complex minimisations are solved with few iterations of a nonlinear solver. The solvers are warmly started with a suitable initial guess, which is extensively discussed, making the algorithm fast. Applications of the algorithm are shown relating to fonts, path planning for human walkers, and as a tool for the time‐optimal lap on a Formula 1 circuit track.