Fourth-order time-stepping for stiff PDEs on the sphere

We present in this paper algorithms for solving stiff PDEs on the unit sphere with spectral accuracy in space and fourth-order accuracy in time. These are based on a variant of the double Fourier sphere method in coefficient space with multiplication matrices that differ from the usual ones, and implicit-explicit time-stepping schemes. Operating in coefficient space with these new matrices allows one to use a sparse direct solver, avoids the coordinate singularity and maintains smoothness at the poles, while implicit-explicit schemes circumvent severe restrictions on the time-steps due to stiffness. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform best. Implementations in MATLAB and Chebfun make it possible to compute the solution of many PDEs to high accuracy in a very convenient fashion

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

Hybrid Chebyshev Polynomial Scheme for the Numerical Solution of Partial Differential Equations

In the numerical solution of partial differential equations (PDEs), it is common to find situations where the best choice is to use more than one method to arrive at an accurate solution. In this dissertation, hybrid Chebyshev polynomial scheme (HCPS) is proposed which is applied in two-step approach and one-step approach. In the two-step approach, first, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then, the resulting homogeneous equation is solved by boundary type methods including the method of fundamental solution (MFS) and the equilibrated collocation Trefftz method. However, this scheme can be applied to solve PDEs with constant coefficients only. So, for solving a wide variety of PDEs, one-step hybrid Chebyshev polynomial scheme is proposed. This approach combines two matrix systems of two-step approach into a single matrix system. The solution is approximated by the sum of particular solution and homogeneous solution. The Laplacian or biharmonic operator is kept on the left hand side and all the other terms are moved to the right hand side and treated as the forcing term. Various boundary value problems governed by the Poisson equation in two and three dimensions are considered for the numerical experiments. HCPS is also applied to solve an inhomogeneous Cauchy-Navier equations of elasticity in two dimensions. Numerical results show that HCPS is direct, easy to implement, and highly accurate

A coupled scheme based on uniform algebraic trigonometric tension B-spline and a hybrid block method for Camassa-Holm and Degasperis-Procesi equations.

[EN]In this article, high temporal and spatial resolution schemes are combined to solve the Camassa-Holm and Degasperis-Procesi equations. The differential quadrature method is strengthened by using modified uniform algebraic trigonometric tension B-splines of order four to transform the partial differential equation (PDE) into a system of ordinary differential equations. Later, this system is solved considering an optimized hybrid block method. The good performance of the proposed strategy is shown through some numerical examples. The stability analysis of the presented method is discussed. This strategy produces a saving of CPU-time as it involves a reduced number of grid points