Finite difference

A finite difference compares the values of a function at finitely separated arguments. For example, the forward finite difference with step $h$ isIt is the discrete analogue of a derivative and lowers the degree of a nonconstant polynomial by one.

Given noisy evaluations at $x_0+h{Z}_i$, where the $Z_i$ are independent Rademacher random variables, the estimatoraverages one-sided finite differences from both directions. If $|g^{\prime \prime }|\le M$ and the noise variance is ${\sigma }^2$, its mean squared error is at most $h^2{M}^2/4+{\sigma }^2/(N{h}^2)$.

A finite difference method replaces derivatives by algebraic combinations of values on a discrete space-time grid.

A nine-point finite-difference stencil on a square grid couples a central value to its four axial neighbours and four diagonal neighbours. Ordering the grid by columns produces a block tridiagonal matrix whose diagonal and off-diagonal blocks are symmetric tridiagonal Toeplitz matrices.

Fourier stability analysis transforms a translation-invariant finite-difference scheme into scalar or matrix recurrences indexed by wavenumber. The Parseval identity then converts uniform multiplier bounds into discrete $2$-norm bounds.

The centered unit-step approximation to a second derivative is

On a uniform grid of spacing $h$, the second-order central differenceapproximates $f^{\prime \prime }(x)$ with error $O(h^2)$ for a sufficiently smooth function.

The forward difference of a sequence or function on the integers is

A discrete antiderivative of $f$ is a function $F$ satisfying $\Delta F=f$. Pascal's identity gives

If ${\Delta }^{k+1}f=0$, then $f$ is an integer linear combination of $1,\left(\genfrac{}{}{0ex}{}{n}{1}\right),\dots ,\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ whenever $f$ is integer-valued. Successive differences recover the coefficients.

Von Neumann analysis inserts the Fourier mode $u_m^n=G^n{e}^{im\theta }$ into a constant-coefficient difference scheme. A one-step method is stable in the discrete $2$-norm when its amplification factor satisfies $|G(\theta )|\le 1$ for every resolvable wavenumber.

A Laurent operator on the whole integer lattice is a translation-invariant bi-infinite matrix. Under the discrete Fourier transform it becomes multiplication by its symbol, so its $2$-operator norm is the essential supremum of the symbol's modulus.

A Toeplitz operator is a constant-diagonal operator on a one-sided sequence space. It models the restriction of a translation-invariant stencil to a half-line, where the boundary prevents direct diagonalization by the full-lattice Fourier transform.

Boundary stability is a mesh-uniform estimate for a finite-difference initial-boundary problem with forced boundary data. Stability of the whole-line Cauchy scheme is necessary but may fail to control boundary-supported numerical modes.

After transforming time and tangential variables, the uniform Kreiss--Lopatinskii condition requires the boundary equations to determine every decaying normal mode with an inverse bounded uniformly over transformed frequencies. It rules out growing or weakly controlled boundary modes.

For a normal amplification matrix $Q$, the bound $\rho (Q)\le 1$ with semisimple unit-modulus eigenvalues controls every power $Q^n$. For a nonnormal family, eigenvalues alone do not control transient growth or give a mesh-uniform bound; eigenvector conditioning, pseudospectra, or a direct energy estimate is then needed.

On a finite inflow grid, explicit upwinding has a triangular amplification matrix whose only eigenvalue is $1-\mu$. For $1<\mu <2$ this eigenvalue lies inside the unit disk, although interior high-frequency data are amplified by $|1-2\mu |>1$ before reaching the boundary. This shows why eigenvalue analysis without uniform normality can give the wrong stability range.

For the centered spatial second difference, backward Euler has amplification factor $G(\theta )=[1+4\mu {\sin }^2(\theta /2){]}^{-1}$ and is stable for every physical Courant number $\mu \ge 0$.

On $J$ interior Dirichlet grid points, backward Euler diffusion is stable for $\mu \ge 0$ and also for the nonphysical branch $\mu \le -[2{\sin }^2(\pi /(2(J+1))){]}^{-1}$. The latter makes every amplification denominator at most $-1$.

For the centered spatial second difference, forward Euler has the updateIt is stable for $0\le \mu \le 1/2$. In this range the update is a convex combination, which gives a direct maximum-norm proof as well as the Fourier proof.

For the schemewhere $L=[1,-2,1]$ is the Dirichlet second-difference matrix and $\mu >0$, the modal amplification factor isMesh-uniform stability holds exactly whenFor consistency with $u_t=u_{xx}$ under the displayed normalization one additionally chooses $a=1/2$; stability then requires $c\le 1/2$.

For a one-step translation-invariant recurrence, the amplification factor is the multiplier satisfyingThe Parseval identity converts the pointwise bound $|H|\le 1$ into non-growth of the discrete $2$-norm.

Forthe Fourier convention $\hat{u}(\theta )={\sum }_m{e}^{-im\theta }{u}_m$ givesprovided the denominator does not vanish.

For a schemethe Fourier ansatz gives the amplification polynomialIts roots are the amplification factors of that Fourier mode.

Centered space and leapfrog time discretization of $u_t+c{u}_x=0$ givesIts amplification polynomial is $G^2+2i\nu \sin \theta G-1$. The two roots have unit modulus for $|\nu \sin \theta |\le 1$, but a repeated unit root at equality violates the uniform root condition.

Applying a centered leapfrog step in time and the centered second difference in space to $u_t=u_{xx}$ givesFor every $\mu >0$, each nonconstant Fourier mode has an amplification root of modulus greater than one, so the scheme is unconditionally unstable.

For the centered spatial second difference, the Crank-Nicolson diffusion scheme has amplification factorIt is stable for every $\mu \ge 0$.

Let $D$ be the real skew-symmetric tridiagonal matrix with superdiagonal $1$ and subdiagonal $-1$. The centered-space Crank-Nicolson discretization of $u_t=u_x$ has amplification matrixThe eigenvalues of $D$ are $2i\cos (j\pi /(M+1))$, so those of $Q$ are Cayley transformsThey all have modulus one, and their common orthonormal eigenbasis makes $Q$ normal. Hence the method is stable for every $\mu >0$.

Forthe amplification polynomial isAll modes have unit-modulus roots only when $\rho =1$ and $0\le \mu \le 1$.

The Courant number is the dimensionless ratio of physical propagation during one time step to one spatial grid spacing, commonly $\mu =c\Delta t/\Delta x$.

A Courant–Friedrichs–Lewy condition requires the numerical domain of dependence to contain the differential equation's physical domain of dependence. It commonly bounds a ratio such as $c\Delta t/\Delta x$ for an explicit hyperbolic scheme.

On $J$ interior points, the one-dimensional centered second-difference matrix with homogeneous Dirichlet boundaries has eigenvalues

If $L$ is the negative semidefinite Dirichlet discrete Laplacian, the Crank-Nicolson amplification matrixhas eigenvalues $(1+\mu {\lambda }_j/2)/(1-\mu {\lambda }_j/2)$ of modulus at most one for every $\mu \ge 0$.

If $T$ is the one-dimensional Dirichlet second-difference matrix on $m$ interior points, the two directional parts of the two-dimensional five-point Laplacian areThey commute, and $A_x+A_y$ is the Kronecker-sum discretization of the Laplacian.

The split stephas amplification matrixOn a common directional eigenvector its eigenvalue is $(1+\mu {\lambda }_p)/(1-\mu {\lambda }_q)$.

On the $m$ by $m$ Dirichlet grid with $h=1/(m+1)$, the exact discrete stability condition for $m>1$ isThe mesh-independent sufficient condition is $0<\mu \le 1/2$, and the exact bound tends to $1/2$ as $h\to 0$.

Adding the correctiongivesIts common-basis eigenvalues areSince ${\lambda }_p,{\lambda }_q<0$, one has $0<d_{pq}\le 1$ for every $\mu >0$.

On a three-dimensional Cartesian grid, the seven-point Dirichlet Laplacian adds the six nearest-neighbour values and subtracts six times the central value. Extending a grid vector by zero on the boundary gives the discrete energy identityso its matrix is symmetric negative definite.

With homogeneous Dirichlet boundary conditions, centered differences turn one-dimensional diffusion into a symmetric negative-definite matrix and constant advection into a skew-symmetric matrix. Their sum dissipates the discrete Euclidean norm, giving mesh-uniform energy stability for every fixed advection coefficient.

A discrete elliptic operator satisfies a discrete maximum principle when a grid function with nonnegative discrete Laplacian cannot have a positive interior maximum unless it is constant. It yields uniqueness and max-norm stability estimates for Dirichlet finite-difference problems.

The method of lines discretizes space in a time-dependent partial differential equation while leaving time continuous, producing a finite system of ordinary differential equations.

For a well-posed linear initial-value problem and a consistent finite-difference approximation, stability is equivalent to convergence.