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As a milestone for compatibility between the numerical stability and accuracy in numerical analysis for advection-diffusion equations, this book presents third-order self-adaptive polynomial schemes to guarantee the nonnegativity condition for any computational conditions in both nonconservative form and conservative form by no means of flux limiters. As an application to advection equations, the present scheme would be suggestive of a necessary and sufficient artificial viscosity to be locally added with the third-order accuracy anywhere. Further this book presents an analytical solution of linear and nonlinear advection-diffusion equations, especially presented a new analytical solution of three-dimensional nonlinear advection-diffusion equations. Based on those analytical solutions, this book presents locally exact noble schemes with nonnegativity properties. These schemes are characterized by a self-consistent nature with an analytical solution of advection-diffusion equations, resulting in guaranteeing the nonnegativity condition anywhere in any steep gradient field. Thus the present locally exact schemes would be a candidate toward "absolutely stable numerical scheme".
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