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In fact, one can write the wave equation as a system of transport equations. Œ0; T Œ0; 1/ solves the wave equation @2t u c2 @2x u D 0. 0; 1/; The existence of a solution can be established via separation of variables. Uniqueness follows from the following conservation principle. Œ0; T Œ0; 1/ solve the initial boundary value problem for the wave equation. x//2 dx for all t 2 Œ0; T. Proof We multiply the partial differential equation by @t u, integrate the resulting identity over x 2 Œ0; 1, and use integration-by-parts to verify that Z 0D 1 0 @2t u @t u c2 @2x u @t u dx D Z 0 1 @2t u @t u C c2 @x u @t @x u dx; where we incorporated the boundary conditions to eliminate boundary terms.

T u Proof Exercise. 4 Explicit Scheme We choose a mesh-size x D 1=J and a step-size t D T=K and replace the partial derivatives in the heat equation by appropriate difference quotients. Ujk /jD0;:::;J at the previous time step. The scheme is also called the explicit Euler scheme. Œ0;T Œ0;1/ for all k D 0; 1; : : : ; K. 1 2 / sup jUjk0 j C 2 j0 D0;:::;J sup jUjk0 j j0 D0;:::;J Ä sup jUjk0 j; j0 D0;:::;J which implies the first assertion. tk ; xj /j: With the estimates for the difference quotients, cf.

Following [6] we will see that a Crank– Nicolson type discretization leads to an implicit scheme that is unconditionally stable and has the same consistency error as the explicit scheme. Ujk 1 /jD0;:::;J are known. The scheme is initialized with the help of an auxiliary time level t 1 as in the case of the explicit scheme. 3 Wave Equation 43 The stability proof mimics the derivation of the energy conservation principle and is based on a discrete version of the integration-by-parts formula. @x Vj / D WJ VJ W0 V0 ; jD1 t u Proof Exercise.