HIGHER-ORDER EVOLUTION EQUATIONS AND SPLIT-STEP METHODS

We mean higher-order equations in time. The most common are $2^{nd}$ order equations. Example:

$$U_{tt}-a^2U_{xx}=0$$

the wave-equation, which admits a solution composed of a right-going and left-going wave. It belongs to the more general 2-order hyperbolic family of equations

$$U_{tt}+2bU_{tx}=a^2U_{xx}+cU_x+dU_t+eU+f(t,x) \mbox {where } b^2<a^2$$

Take $\left\{\begin{array}{l} U_{tt}-a^2 U_{xx}=0\\ U(0,x)=U_0(x)\\ U_t(0,x)=U_1(x) \end{array}\right.$

has a general solution

$$U(x,t)=\frac12\Big[U_0(x-at)+U_0(x+a_t)\Big]+\frac{1}{2a}\int^{x+at} _{x-at}U_1(y)dy$$

As you can see, the solution is composed of a left-going and right-going wave. The above problem can also be cast as a system of equations:

Example

On $-1\le x\le 1$     $t\ge 0$

Take      $U_{tt}=U_{xx}$. It is equivalent to solving

$\left\{\begin{array}{ll} U_t+V_x=0 &\\ V_t+U_x=0 & \mbox {on } 0\le x\le 1,\quad t\ge 0 \end{array}\right.$

So we can let ${\bf V}=(u,v)^T$ and solve

$$\frac{\partial {\bf V}}{\partial t}+A\frac{\partial {\bf V}}{\partial x}={\bf0}$$   $$\mbox{ where $A$ is diagonalizable and has only real e'values}$$

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Example Another higher-order equation which appears with some regularity: The Euler-Bernouilli Equation $U_{tt}=-b^2 U_{xxxx}$ also has a general solution composed of 2 basic solutions. This equation is neither parabolic or hyperbolic $\ldots$ the solution does not become smoother as $t\rightarrow\infty$, like parabolic equations, nor does the solution have finite speed of propagation as it does for hyperbolic equations.

$\Box$

We will not consider in detail the general solution of higher-in-time PDE's. Merely, we indicate the general technique for their solution. A sensible technique is to turn the 2nd-order (or higher order) equation into a system of $1^{st}$-order equations, and then, after projecting or discretizing in space, use ODE theory to find the best time integration algorithm for the resulting problem. Caution: Make sure that you engineer the eventual linear-algebraic problem in a compact way. If you use the above trick you will get very large and sparse matrices with large bandwidths. You can most likely turn the sparse large band width matrix problem into a sparse small bandwidth problem using careful computer engineering practices. We'll see how this is done in a number of examples considered in this section and in the Elliptic Equation Section.