# Time Domain Room Acoustic Solver with Fourth-Order Explicit FEM Using Modified Time Integration
## 3 New Room Acoustic Solver Using Explicit TD-FEM with Modified Adams Method
### Designing a Higher-Order Time Integration Method
#### 3.2.3 Modified Fourth-Order Adams-Bashforth Method  
To achieve the fourth-order accuracy in the resulting time marching scheme, we design appropriate weight coefficients of Adams-Bashforth method using time domain dispersion error analysis. With \(l=4\), the linear multi-step form without the source and the dissipation terms for dispersion error analysis is expressed as  
\[(\mathbf{p}^{n+1}-2\mathbf{p}^{n}+\mathbf{p}^{n-1})+\Delta t^{2}c_{0}^{2}\mathbf{D}^{-1}\mathbf{ M}\mathbf{D}^{-1}\mathbf{K}(b_{1}\mathbf{p}^{n}+b_{2}\mathbf{p}^{n-1}+b_{3}\mathbf{p}^{n-2}+b_{4} \mathbf{p}^{n-3})=0. \tag{28}\]  
Here, the same two-dimensional free field used in frequency domain dispersion analysis (Section 3.1) is assumed, but the plane wave in time domain is defined as  
\[p_{x,y}^{n}=e^{i(kx\cos\!\theta+ky\sin\!\theta-\omega n\Delta t)}. \tag{29}\]  
Using Equation (29), Equation (28) is transformed into  
\[2(\cos\!\omega\Delta t-1)\mathbf{p}_{x,y}^{n}+\Delta t^{2}c_{0}^{2}A(b_{1}+b_{2}e ^{i\omega\Delta t}+b_{3}e^{2i\omega\Delta t}+b_{4}e^{3i\omega\Delta t})\mathbf{p}_ {x,y}^{n}=0. \tag{30}\]  
From Equation (30), the dispersion relation is represented as  
\[c_{0}=\sqrt{\frac{(1-\cos\!c^{h}\!k^{h}\Delta t)}{A\Delta t^{2}(b_{1}+b_{2}e ^{ic^{h}\!k^{h}\Delta t}+b_{3}e^{2 ic^{h}\!k^{h}\Delta t}+b_{4}e^{3 ic^{h}\!k^{ h}\Delta t})}}. \tag{31}\]By taking Taylor expansion of \(k^{h}\) in Equation (31), \(c^{h}\) is evaluated as  
\[c^{h}\approx c_{0}(1+i\frac{(kc_{0}\Delta t)}{2}X_{1}+\frac{(kc_{0}\Delta t)^{2} }{24}X_{2}-\frac{i(kc_{0}\Delta t)^{3}}{48}X_{3}+\mathcal{O}(k^{4})), \tag{32}\]  
with  
\[\begin{split} X_{1}&=b_{2}+2b_{3}+3b_{4},\\ X_{2}&=(b_{1}-2b_{2})^{2}-22b_{1}b_{3}+8b_{2}b_{3}+ 13b_{3}^{3}-52b_{1}b_{4}-4b_{2}b_{4}+32b_{3}b_{4}+28b_{4}^{2},\\ X_{3}&=b_{1}^{2}(5b_{2}+34b_{3}+111b_{4})+b_{2}^{2}(-8 b_{1}+2b_{2}+8b_{3}+22b_{4})+b_{3}^{2}(-76b_{1}+b_{2}+10b_{3}+35b_{4})\\ &+b_{4}^{2}(-264b_{1}-70b_{2}+40b_{3}+30b_{4})-30b_{1}b_{2}b_{3}- 24b_{2}b_{3}b_{4}-250b_{3}b_{4}b_{1}-16b_{4}b_{1}b_{2}.\end{split} \tag{33}\]  
By eliminating the first-order to the third-order dispersion error terms in Equation (32) while satisfying Equation (25), we obtain the following modified weight coefficients as  
\[b_{1}=14/12,\ b_{2}=-5/12,\ b_{3}=4/12,\ b_{4}=-1/12. \tag{34}\]  
With the coefficients in Equation (34), \(c^{h}\) is defined as  
\[c^{h}\approx c_{0}(1-\frac{(kh)^{4}}{1440}(57\tau^{4}+8-19\cos^{2}\!\theta\sin ^{2}\!\theta)-i\frac{(\omega\Delta t)^{5}}{24}). \tag{35}\]  
Equation (35) shows clearly that by using the designed Adams-Bashforth method with the modified weight coefficients of Equation (34), the resulting time marching scheme has fourth-order accuracy in both space and time. In addition, dispersion error analysis shows that the resulting scheme includes the fifth-order dissipation error term, but its magnitude is small; moreover, it is controllable in a simple manner. The effect of dissipation error is evaluated theoretically and numerically in Section 4, including the presentation of a simple control method. With the modified weight coefficients of Equation (34) the complete expression of time marching scheme of sound pressure in Equation (24) is  
\[\mathbf{p}^{n}=\mathbf{p}^{n-1}+\frac{\Delta t}{12}\mathbf{D}^{-1}(14\mathbf{M}v^{n-1}-5\mathbf{M }v^{n-2}+4\mathbf{M}v^{n-3}-\mathbf{M}v^{n-4}). \tag{36}\]  
The proposed explicit TD-FEM using the modified Adams method comprises Equations (36) and (8); it also uses the modified integration points of Equation (21) in element matrices calculation.