此方法目的是提供一種質量體積守恆的方法，適用於同時擁有對流及擴散影響之方程式。
在對流影響部分，使用片段連續的常數函數做為有限元素法之基底函數藉以近似原方程式之真解；
在擴散影響部分，使用有限差分法來處理來達到近似的目的。其中使用混合的方法將對流及擴散兩部分連接起來藉以達到接近真解。

The goal of this method is to implement the volume and mass
conserving characteristic method. This method is appropriate to the
equation with advections and diffusions. Characteristic mixed FEM
with piecewise constant approximation is applied to the advection
part, and the diffusion part is handled by FDM.

1 Introduction 5
2 Methods 7
2.1 The advection part . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Boundary Treatment . . . . . . . . . . . . . . . . . . . 10
2.2 For the diffusion part . . . . . . . . . . . . . . . . . . . . . . . 12
3 Numerical Diffusion 14
4 Numerical Experiment 19
5 Conclusion 29
References

[1] Michael A. Celia, Thomas F. Russell, Ismael Herrera and Richard E. Ewing, An Eulerian-Lagrangian localized adjoint method fo the advectiondiffusion equation, Adv. Water Resources, 1990, Vol. 13, No. 4, (1990) 187-206.
[2] Jim Douglas, JR. and Chieh-Sen Huang, A locally conservative Eulerian–Lagrangian Finite Difference Method for a parabolic equation, BIT, Vol. 41, No. 3, (2001) 480-489.
[3] Strikwerda J.C, Finite Difference Schemes and Partial Differential Equations, Chapman and Hall, New York, (1989).
[4] Todd arbogast and C.S. Huang, A fully mass and volume conserving implementation of a characteristic method for transport problems, SIAM Journal on scientific Computing, Vol 28, Issue 6 2001-2022. (2006)
[5] Jim Douglas, Jr., Chieh-Sen Huang, Felipe Pereira, The modified method of characteristics with adjusted advection Numer. Math. 83: 353–369 (1999).
[6] HongWang, Richard E. Ewing, Thomas F. Russell, Eulerian-Lagrangian Localized Adjoint Methods for Convection-Diffusion Equations and Their Convergence Analysis, IMA J. Numer. Anal., 15, 405-459 (1995).
[7] Todd Arbogast, Mary F Wheeler, A Characteristics-Mixed Finite Element Method for Advection-Dominated Transport Problems, SIAMJournal on Numerical Analysis, Vol. 32, 404-424, (1995).