In this paper we used the method of parabola approximation to study some nonlinear differential equations. We derive exact, explicit solutions to the parabolic equations and use this analytical results in the numerical computations for the general equations. We then draw the comparison of between the solutions of original and approximated equations. Moreover, we apply such method to the population growth problem. The error of the difference between the solutions of the differential equations and the numerical results caused by the discrete approximations is reasonable.

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In this study, partition of the energy equation in the Euler equations is expanded in the form of ideal mixture gas through the finite-volume method in order to discuss questions faced in the Riemann approximation solver. Through the AUSMDV numerical flux scheme with different flux limiters and types of gradient, we want to improve the calculation accuracy and reduce the nonphysical oscillation problem. Based on the AUSMDV scheme, We modify the energy equation of the ideal mixture gas which was proposed by Ton [8]. For the pressure and velocity gradient of source term Q, we establish a new method of gradient control to test whether it can make progress in the nonphysical numerical oscillation problem.

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