Linearly implicit generalized trapezoidal formulas for nonlinear differential equations

Abstract

We describe a linearlized version of the class of generalized trapezoidal formulas (GTFs) introduced in Chawla et al.[3]. For nonlinear differential equations, the obtained one-parameter class of linearly implicit generalized trapezoidal formulas (LIGTF(α)) obviate the need to solve a nonlinear system at each time step of integration, while they retain the order of accuracy and stability properties of the (functionally implicit) GTFs. The performance of the present LIGTF(α) is compared with the linearized linearly implicit trapezoidal formula (Lintrap) for nonlinear stiff ordinary differential equations (ODEs), and for nonlinear partial differential equations (PDEs) which represent nonlinear transportation-diffusion, nonlinear diffusion and nonlinear reaction-diffusion. Lintrap is known to produce unwanted oscilliations if the ratio of the time step to the spatial step becomes large. In our numerical experiments, the significance of the role played by the parameter in LIGTF(α) becomes evident in providing both stability and accuracy of the computed solution in the presence of diffusivity.

Keywords:

• Nonlinear differential equations
• finite-difference schemes
• linearly implicit generalized trapezoidal formulas
• stiff systems
• nonlinear transportation-diffusion
• viscous Burger's equation
• nonlinear diffusion
• nonlinear reaction-diffusion

C.R. Category:

• G.1.8