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By means of easy examples, such as the Korteweg-de Vries, the Harry Dym, the sine-Gordon equations, and the Hirota coupled system, it is shown how nonlinear partial differential equations can be exactly solved by a direct algebraic method. The physical concept, on which the method relies, is one of generation and mixing of the real exponential solutions of the underlying linear equations. This approach leads in a straightforward way to single solitary waves of pulse, kink and cusp shape. The extension of the method towards the construction of multi-soliton solutions and the connections with other direct methods are outlined.

By means of easy examples, such as the Korteweg-de Vries, the Harry Dym, the sine-Gordon equations, and the Hirota coupled system, it is shown how nonlinear partial differential equations can be exactly solved by a direct algebraic method. The physical concept, on which the method relies, is one of generation and mixing of the real exponential solutions of the underlying linear equations. This approach leads in a straightforward way to single solitary waves of pulse, kink and cusp shape. The extension of the method towards the construction of multi-soliton solutions and the connections with other direct methods are outlined.

In this work, we construct novel discretizations for the unsteady convection-diffusion equation. Our discretization relies on multiderivative time integrators together with a novel discretization that reduces the total number of unknowns for the solver. These type of temporal discretizations come from an umbrella class of methods that include Lax-Wendroff (Taylor) as well as Runge-Kutta methods as special cases. We include two-point collocation methods with multiple time derivatives as well as a sixth-order fully implicit collocation method that only requires a total of three stages. Numerical results for a number of sample linear problems indicate the expected order of accuracy and indicate we can take arbitrarily large time steps.

A novel explicit and implicit Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) scheme is presented for hyperbolic equations such as Burgers equation and compressible Euler equations. The proposed scheme performs better than the original SUPG stabilized method in multi-dimensions. To demonstrate the numerical accuracy of the scheme, various numerical experiments have been carried out for 1D and 2D Burgers equation as well as for 1D and 2D Euler equations using Q4 and T3 elements. Furthermore, spectral stability analysis is done for the explicit 2D formulation. Finally, a comparison is made between explicit and implicit versions of the KSUPG scheme.