A modified finite volume method for convection-diffusion-reaction problems

- A modified finite volume method is developed to solve the convection-diffusion-reaction problem.
- The MFVM is much more accurate and stable than the traditional FVM.
- The modified finite volume method sometime can achieve almost the same accuracy as the exact solution of the convection-diffusion-reaction problem.
- The modified finite volume method is as simple as the traditional finite volume method.

Usually for the conventional numerical methods the increase of their accuracy and stability is at the cost of increasing the size of the computational stencil. In the present work, without widening the computational stencil of the classical finite volume method (FVM) we propose a modified finite volume method (MFVM) which is much more accurate and stable than the FVM. In the MFVM some undetermined coefficients are introduced in discretizing the convection-diffusion-reaction equation. These coefficients are determined analytically by making the truncation error of the numerical scheme vanish and by expressing the high order derivatives of the unknown function with low order ones through the governing equation. Interestingly, for some one-dimensional convection-diffusion-reaction problems the MFVM can achieve almost the same accuracy as the exact solution, and does not induce any unphysical oscillations for the convection/reaction-dominated problems.