Method of lines

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 periodic indexing let $S$ be the unitary forward shift. The operator is $A=-3I/2+2S-S^2/2$. A discrete Fourier transform diagonalizes it with eigenvalue $\lambda (\theta )=-3/2+2e^{i\theta }-e^{2i\theta }/2$ and real part $-(1-\cos \theta {)}^2$. Consequently $\Vert e^{tA/h}{\Vert }_2\le 1$ for every $t\ge 0$, uniformly in the grid. Equivalently $A+A^{\ast }=-\frac{1}{2}(2I-S-S^{\ast }{)}^2$, proving decay of the discrete energy directly. The result is semidiscrete stability; a further time discretization must also have adequate absolute stability.

For homogeneous Dirichlet endpoints, the centered first-difference matrix is skew-Hermitian and the centered second-difference matrix is negative definite. Thus $\dot{U}=(D_{xx}-\alpha D_x)U$ is contractive in the discrete L2 norm for every real $\alpha$ and every mesh width. The displayed identity follows by summation by parts. Nonnegative off-diagonal entries additionally require $|\alpha |h\le 2$; that maximum-principle condition is not necessary for the L2 energy estimate. numerical consistency and Duhamel's formula then give second-order numerical convergence for smooth solutions and compatible initial approximations.

Let $K_h$ be a symmetric nonnegative matrix and consider $U^{\prime \prime }=(\alpha I-K_h)U$ with fixed real $\alpha$. In any inner product for which $K_h$ is self-adjoint, diagonalization of a matrix givesIndeed each modal coefficient is either a cosine and sine divided by its frequency, a linear function at zero frequency, or a hyperbolic cosine and sine divided by its growth rate. The bounds $|\cos (\omega t)|\le 1$, $|\sin (\omega t)/\omega |\le t$, and $\sinh (\beta t)/\beta \le t{e}^{\beta t}$ prove the estimate. It is uniform in the spectrum of $K_h$ and therefore gives finite-time displacement stability. Uniform velocity estimates need additional control of the initial spatial energy. All-time displacement bounds require a strictly positive lower bound for $K_h-\alpha I$, a different condition.

For $U^{\prime \prime }=(\alpha I-K_h)U$ with symmetric positive definite $K_h$, all-data displacement is bounded for all time on a fixed grid exactly when $\alpha <{\lambda }_{\min }(K_h)$. Diagonalization of a matrix reduces this assertion to oscillators. At equality, an initial velocity in a zero-frequency mode produces linear growth; above it a growing hyperbolic mode is available. A lower bound on ${\lambda }_{\min }(K_h)-\alpha$ uniform in the mesh gives a uniform displacement estimate. This all-time statement is stronger than displacement stability of a symmetric semidiscrete wave equation on fixed finite time intervals.

If a spatial method of lines has a Hermitian matrix $H$, its generator $-iH$ is a skew-Hermitian matrix. The matrix exponential $e^{-itH}$ is a unitary matrix, so it preserves the Euclidean norm and any constant-volume discrete L2 norm. Equivalently, differentiating the squared norm gives $2\mathrm{Re}(-i{U}^{\ast }HU)=0$. A real sampled potential preserves the Hermitian property of a symmetric discrete Laplacian. Subsequent time discretization must be assessed separately.