Energy method

The energy method differentiates a norm or quadratic energy along a differential equation or numerical evolution and estimates the resulting terms. Dissipative, skew-adjoint, and boundary contributions can yield stability or conservation without diagonalizing the operator.

For zero endpoint values and real constant drift, integration by parts gives $\frac{1}{2}d\Vert u{\Vert }_{L^2}^2/dt=-\Vert u_x{\Vert }_{L^2}^2$. The drift contributes only a zero boundary term in the real part. Thus the solution is contractive, uniformly in the drift. Multiplication by $e^{\kappa x}$ converts the equation to a heat equation with constant killing rate ${\kappa }^2$, providing existence and uniqueness as well as the direct energy estimate.

The semidiscrete system $U^{\prime }=D_{xx}U+\alpha D_{0}U$, for real fixed $\alpha$, is contractive in the periodic discrete L2 norm. Discrete summation by parts makes the drift term's real energy contribution zero and the diffusion term nonpositive. Equivalently its Fourier amplification symbol generator has real part $-4h^{-2}{\sin }^2(\theta /2)$. This continuous-time estimate does not prove stability of an arbitrarily chosen time integrator.

For the centered second difference and centered first difference on a finite interval with zero Dirichlet boundary conditions, the diffusion matrix is symmetric negative definite and the drift matrix is skew-symmetric. Discrete summation by parts gives the displayed mesh-weighted identity. It proves contraction in the discrete L2 norm without needing periodic modes, positive stencil coefficients or simultaneous diagonalization.

For a zero-boundary centered semidiscretization $U^{\prime }=D_{h}U+V_{h}U$, where $D_h$ is the scaled Dirichlet discrete Laplacian and $V_h=\mathrm{diag}(a_j)$ with $a_j\le a_{+}$, summation by parts gives $(U,D_{h}U{)}_h\le 0$. Hence $\frac{1}{2}(d/dt)\Vert U{\Vert }_h^2\le a_{+}\Vert U{\Vert }_h^2$. The Gronwall inequality yields $\Vert U(t){\Vert }_h\le e^{a_{+}t}\Vert U(0){\Vert }_h$, a mesh-independent stability bound on every fixed finite interval. Positive reaction coefficients can permit physical growth without violating this form of stability.

For the weak energy solution of a variable-coefficient wave equation, uniqueness can be proved without using $u_t$ as a spatial $H_0^1$ test function. For a difference with zero data, fix $s$ and test by $v(t)={\int }_t^{s}u(r)dr$ for $t<s$, extended by zero afterwards. Put $z(t)={\int }_0^{t}u(r)dr$, so $v(t)=z(s)-z(t)$. Integration by parts in time for the symmetric principal form, and in space for the first-order terms, giveFor short enough intervals the last term is absorbed. The Gronwall inequality forces $u=0$, and repetition proves uniqueness on the full interval. This antiderivative test is valid at the stated space-time $H^1$ regularity, unlike an unqualified direct energy test by $u_t$.