  In this paper, we propose an accurate finite difference method to discretize
the $d$-dimensional (for $d\ge 1$) tempered integral fractional Laplacian and
apply it to study the tempered effects on the solution of problems arising in
various applications. Compared to other existing methods, our method has higher
accuracy and simpler implementation. Our numerical method has an accuracy of
$O(h^\epsilon)$, for $u \in C^{0, \alpha+\epsilon} (\bar{\Omega})$ if $\alpha <
1$ (or $u \in C^{1, \alpha-1+\epsilon} (\bar{\Omega})$ if $\alpha \ge 1$) with
$\epsilon > 0$, suggesting the minimum consistency conditions. The accuracy can
be improved to $O(h^2)$, for $u \in C^{2, \alpha+\epsilon} (\bar{\Omega})$ if
$\alpha < 1$ (or $u \in C^{3, \alpha - 1 + \epsilon} (\bar{\Omega})$ if $\alpha
\ge 1$). Numerical experiments confirm our analytical results and provide
insights in solving the tempered fractional Poisson problem. It suggests that
to achieve the second order of accuracy, our method only requires the solution
$u \in C^{1,1}(\bar{\Omega})$ for any $0<\alpha<2$. Moreover, if the solution
of tempered fractional Poisson problems satisfies $u \in C^{p,
s}(\bar{\Omega})$ for $p = 0, 1$ and $0<s \le 1$, our method has the accuracy
of $O(h^{p+s})$. Since our method yields a (multilevel) Toeplitz stiffness
matrix, one can design fast algorithms via the fast Fourier transform for
efficient simulations. Finally, we apply it together with fast algorithms to
study the tempered effects on the solutions of various tempered fractional
PDEs, including the Allen-Cahn equation and Gray-Scott equations.
